ALBERT

Description: The aim of this paper is to study the relations between the Hausdorff dimensions of k -quasilines and the theory of extremal quasiconformal mappings. We show that there is an open and dense subset (Strebel points) of the universal Teichmüller space T (H) such that, for every [ f ] in the set, the Hausdorff dimension of the k -quasiline determined by [ f ] is strictly less than 1 + k 2 . We also show that there are some points [ f ] != [id] outside the open and dense set in the universal Teichmüller space such that the Hausdorff dimension of the quasiline determined by [ f ] is 1. Moreover, some results on the Hausdorff dimensions of the quasilines varying in the asymptotic Teichmüller space are also obtained.

Description: A new uniqueness result for a general n th order differential equation is obtained. We show that some previous results follow immediately from our theorem.

Description: Consider a discrete uniformly elliptic divergence form equation on the d ≥3 dimensional lattice Z d with random coefficients. In Conlon and Spencer [ Trans. Amer. Math. Soc. , http://www.math.lsa.umich.edu/~conlon/paper/hom10.pdf ], rate of convergence results in homogenization and estimates on the difference between the averaged Green's function and the homogenized Green's function for random environments which satisfy a Poincaré inequality were obtained. Here, these results are extended to certain environments in which correlations can have arbitrarily small power law decay. These environments are simply related via a convolution to environments which do satisfy a Poincaré inequality.

Description: In this paper, we prove that, for any a , M N with ( a , M ) = 1, there are infinitely many Carmichael numbers m such that m a mod M .

Description: We give a topological analogue for openness of a criterion for flatness that originates with Auslander. Over a normal base of dimension n , failure of openness is detected by a vertical component in the n th fibred power of the morphism.

Description: Investigated are regular maps from real algebraic varieties into real Fermat varieties. It is proved that under some natural assumptions, all such maps are null homotopic.

Description: Let ( M , g ) be a complete non-compact Riemannian manifold. We consider operators of the form g + V , where g is the non-negative Laplacian associated with the metric g , and V a locally integrable function. Let be a Riemannian covering, with Laplacian g and potential . If the operator + V is non-negative on ( M , g ), then the operator is non-negative on . In this note, we show that the converse statement is true provided that is a co-amenable subgroup of 1 ( M ).

Description: Peak interpolation is concerned with a foundational kind of mathematical task: building functions in a fixed algebra A , which have prescribed values or behaviour on a fixed closed subset (or on several disjoint subsets). In this paper, we do the same but now A is an algebra of operators on a Hilbert space. We briefly survey this noncommutative peak interpolation , which we have studied with coauthors in a long series of papers, and whose basic theory now appears to be approaching its culmination. This programme developed from, and is based partly on, theorems of Hay and Read whose proofs were spectacular, but therefore inaccessible to an uncommitted reader. We give short proofs of these results, using recent progress in noncommutative peak interpolation, and conversely give examples of the use of these theorems in peak interpolation. For example, we prove a useful new noncommutative peak interpolation theorem.

Description: We show how a parameterized family of maps of the spine of a manifold can be used to construct a family of homeomorphisms of the ambient manifold which have the inverse limits of the spine maps as global attractors. We describe applications to unimodal families of interval maps, to rotation sets, and to the standard family of circle maps.

Description: For every prime p , we give infinitely many examples of torsors under abelian varieties over Q that are locally trivial but not divisible by p in the Weil–Châtelet group. We also give an example of a locally trivial torsor under an elliptic curve over Q that is not divisible by 4 in the Weil–Châtelet group. This gives a negative answer to a question of Cassels.

Description: We show that at least of the zeros of the Riemann zeta-function are simple, assuming the Riemann hypothesis. This was previously established by Conrey, Ghosh and Gonek [ Proc. London Math. Soc. 76 (1998) 497–522] under the additional assumption of the generalized Lindelöf hypothesis. We are able to remove this hypothesis by careful use of the generalized Vaughan identity.

Description: We construct an infinite family of hyperbolic, homologically thin knots that are not quasi-alternating. To establish the latter, we argue that the branched double-cover of each knot in the family does not bound a negative-definite 4-manifold with trivial first homology and bounded second Betti number. This fact depends in turn on information from the correction terms in Heegaard Floer homology, which we establish by way of a relationship to, and calculation of, the Turaev torsion.

Description: We observe that E -resultant of a very ample rank 2 vector bundle E on a real projective curve (with no real points) is nonnegative when restricted to the space of real sections. Moreover, we show that if E has a section vanishing at exactly two points and the degree d of E satisfies d ( d – 6) ≥ 4( g – 1), then this polynomial cannot be written as a sum of squares.

Description: Let G be one of the Ricci-flat holonomy groups SU( n ), Sp( n ), Spin(7) or G 2 , and M a compact manifold of dimension 2 n , 4 n , 8 or 7, respectively. We prove that the natural map from the moduli space of torsion-free G -structures on M to the moduli space of Ricci-flat metrics is open, and that the image is a smooth manifold. For the exceptional cases G = Spin(7) and G 2 , we extend the result to asymptotically cylindrical manifolds.

Description: We improve the results of Booker and Krishnamurthy ( Compos. Math. 147 (2011) 669–715) by allowing restricted sets of poles among the unramified twists. This allows for a clean statement of the GL(2) converse theorem which includes all cases of Eisenstein series.

Description: Let G be a solvable subgroup of the automorphism group Aut( X ) of a compact Kähler manifold X of complex dimension n , and let N ( G ) be the normal subgroup of G consisting of elements with null entropy. Let us denote by G * the image of G under the natural map from Aut( X ) to GL( V , R ), where V is the Dolbeault cohomology group H 1, 1 ( X , R ). Assume that the Zariski closure of G * in GL( V C ) is connected. The main aim of this paper is to show that, when the rank r ( G ) of the quotient group G / N ( G ) is equal to n – 1 and the identity component of Aut( X ) is trivial, then the normal subgroup N ( G ) of G is finite. This affirmatively answers a question in Invent. Math. posed by D.-Q. Zhang.

Description: The dihedral angle formed at junctions between two plagioclase grains and a grain of augite is only very rarely in textural equilibrium in gabbros from kilometre-scale crustal layered intrusions. The median of a population of these disequilibrium angles, cpp , varies systematically within a single layered intrusion, remaining constant over large stretches of stratigraphy with significant increases and decreases associated with the addition or reduction respectively of the number of phases on the liquidus of the bulk magma. The stepwise changes in cpp are present in the Upper Zone of the Bushveld Complex, the Megacyclic Unit I of the Sept Iles Intrusion, and the Layered Series of the Skaergaard intrusion. The plagioclase-bearing cumulates of Rum have a bimodal distribution of cpp , dependent on whether the cumulus assemblage includes clinopyroxene. The presence of the stepwise changes is independent of the order of arrival of cumulus phases and of the composition of either the cumulus phases or the inferred composition of the interstitial liquid. The only parameter that behaves in an exactly analogous manner to cpp is the rate of change in enthalpy with temperature ( H / T ) during crystallization. Both H / T and cpp increase with the addition of a liquidus phase, and decrease with the removal of a liquidus phase. The replacement of one phase by another has little effect on H / T and no discernible effect on cpp . An increase of H / T results in an increase in the fraction of the total enthalpy budget that is the latent heat of crystallization (the fractional latent heat). It also increases the mass crystallized in each incremental temperature drop (the crystal productivity). These increases of both fractional latent heat and crystal productivity are likely to cause an increase in the time taken to form three-grain junctions in the mush via thermal buffering of a thickened mushy layer. We suggest these are the underlying causes of stepwise increases in cpp . Stepwise changes in the geometry of three-grain junctions in fully solidified gabbros thus provide a clear microstructural marker for the progress of fractionation down the liquid line of descent in layered intrusions.

Description: High-resolution sampling in monogenetic fields has the potential to reveal fine-scale heterogeneity of the mantle, a feature that may be overwhelmed by larger fluxes of magma, or missed by under-sampling. The Quaternary Auckland Volcanic Field (AVF) in northern New Zealand is a basaltic field of 51 small-volume volcanic centres, and is one of the best-sampled examples of a monogenetic volcanic field. We present data for 12 centres in the volcanic field. These show the large compositional variations between volcanoes as well as through single eruptive sequences. Whole-rock compositions range from subalkaline basalt in the larger centres, through alkali basalt to nephelinite in the smallest centres. Fractional crystallization has had a limited effect in many of the centres, but high-pressure clinopyroxene crystallization may have occurred in others. Three end-members are observed in Pb isotope space, indicating that distinct mantle source components are involved in the petrogenesis of the magmas. Whole-rock multi-element patterns show that the larger centres have prominent positive Sr anomalies and lack K anomalies, whereas the smaller centres have prominent negative K anomalies and lack Sr anomalies. The melting parameters and compositions of the sources involved are modelled using trace element ratios and multi-element patterns, and three components are characterized: (1) fertile peridotite with a Pb-isotope composition similar to Pacific mid-ocean ridge basalt; (2) eclogite domains with a HIMU-like isotope composition dispersed within the fertile peridotite; (3) slightly depleted subduction-metasomatized peridotitic lithospheric mantle (containing c . 3% subduction fluids). Modelling shows that melting in the AVF begins in garnet-bearing fertile asthenosphere (with preferential melting of eclogite domains) and that melts are variably diluted by melts of the lithospheric source. The U–Th isotope compositions of the end-members in the AVF show 230 Th excess [( 230 Th/ 232 Th) ratios of 1·11–1·38], with the samples of lower ( 230 Th/ 232 Th) exhibiting higher ( 238 U/ 232 Th), which we attribute to the dilution effect of the melts from the lithospheric mantle source. Modelling reveals a correlation between melting in the asthenosphere, the degree of melting and incorporation of the metasomatized lithospheric mantle source, and the resultant size of the volcanic centre. This suggests that the scale of the eruption may essentially be controlled by asthenospheric mantle dynamics.

Description: The origin of mafic and ultramafic sills exhibiting different whole-rock compositional profiles (e.g. I-, C-, D-, M- and S-shaped profiles) remains controversial. We have addressed this issue by revisiting three ~100 m thick Siberian dolerite sills (Vavukansky, Kuz’movsky and Vilyuysky) that display remarkable internal differentiation. The Vavukansky sill has an M-shaped profile with prominent basal and top reversals showing inward increases in whole-rock MgO, Mg-number [100Mg/(Mg + Fe)] and normative An content [100An/(An + Ab)], followed by the Layered and Upper Border Series with inward decreases in these indices. The Kuz’movsky and Vilyuysky sills both show S-shaped profiles similar to the Vavukansky sill, but lack a top reversal. These whole-rock M- and S-shaped profiles are accompanied by similar profiles in mineral compositions. Plagioclase and, to a lesser extent, olivine show systematic inward increases in An content and Mg-number, respectively, across basal and top reversals. These compositional trends are followed by inward decreases in these ratios in the interiors of the Vavukansky and Kuz’movsky sills. Currently accepted models attribute whole-rock M- and S-shaped compositional profiles to crystal settling, compositional convection or compaction operating in closed systems. Our observations challenge these traditional interpretations because variations in mineral compositions observed in marginal reversals cannot result from closed-system fractionation. We suggest instead that initially the sills evolved as open systems that were slowly inflated by magmas that became gradually more primitive with time. The inflation was accompanied by in situ crystallization that preserved the preceding fractionation history of the injected magmas by forming basal and top reversals with minerals becoming more primitive inwards. This process culminated with rapid inflation of the sills to their current size owing to a major influx of primitive magma. Subsequently, magma flow through the sills ceased and they evolved as closed systems by fractional crystallization. This resulted in the Layered and Upper Border Series with minerals becoming more evolved inwards. This model can be extended to explain other compositional profiles and petrological features in mafic and ultramafic sills.

Description: We show the existence and uniqueness of a solution for the nonlocal vector-valued Allen–Cahn variational inequality in a formulation involving Lagrange multipliers for local and nonlocal constraints. Furthermore, we propose and analyse a primal–dual active set (PDAS) method for local and nonlocal vector-valued Allen–Cahn variational inequalities. The local convergence behaviour of the PDAS algorithm is studied by interpreting the approach as a semismooth Newton method and numerical simulations are presented demonstrating its efficiency.

Description: In this paper, we will present a generalized convolution quadrature for solving linear parabolic and hyperbolic evolution equations. The original convolution quadrature method by Lubich works very nicely for equidistant time steps while the generalization of the method and its analysis to nonuniform time stepping is by no means obvious. We will introduce the generalized convolution quadrature allowing for variable time steps and develop a theory for its error analysis. This method opens the door for further development towards adaptive time stepping for evolution equations. As the main application of our new theory, we will consider the wave equation in exterior domains which is formulated as a retarded boundary integral equation.

Description: We address the error control of Galerkin discretization (in space) of linear second-order hyperbolic problems. More specifically, we derive a posteriori error bounds in the L ( L 2 ) norm for finite element methods for the linear wave equation, under minimal regularity assumptions. The theory is developed for both the space-discrete case and for an implicit fully discrete scheme. The derivation of these bounds relies crucially on carefully constructed space and time reconstructions of the discrete numerical solutions, in conjunction with a technique introduced by Baker (1976, Error estimates for finite element methods for second-order hyperbolic equations. SIAM J. Numer. Anal. , 13 , 564–576) in the context of a priori error analysis of Galerkin discretization of the wave problem in weaker-than-energy spatial norms.

Description: The surface finite element method can be used to approximate curvatures on embedded hypersurfaces and to discretize geometric partial differential equations. In this paper, we present a definition of discrete Ricci curvature on polyhedral hypersurfaces of arbitrary dimension based on the discretization of a weak formulation with isoparametric finite elements. We prove that for a piecewise quadratic approximation of a two- or three-dimensional hypersurface R n +1 , this definition approximates the Ricci curvature of with a linear order of convergence in the L 2 ( ) norm. By using a smoothing scheme in the case of a piecewise linear approximation of , we still get a convergence of order 2/3 in the L 2 ( ) norm and of order 1/3 in the W 1, 2 ( ) norm.

Description: We give general conditions which guarantee that the sequence generated by a descent algorithm converges to an equilibrium point. The convergence result is based on the Lojasiewicz gradient inequality; optimal convergence rates are also derived, as well as a stability result. We show how our results apply to a large variety of standard time discretizations of gradient-like flows. Schemes with variable time step are considered and optimal conditions on the maximal step size are derived. Applications to time and space discretizations of the Allen–Cahn equation, the sine–Gordon equation and a damped wave equation are given.

Description: This paper presents quadratic finite-volume methods for elliptic and parabolic problems on quadrilateral meshes that use Barlow points (optimal stress points) for dual partitions. Introducing Barlow points into the finite-volume formulations results in better approximation properties at the cost of loss of symmetry. The novel ‘symmetrization’ technique adopted in this paper allows us to derive optimal-order error estimates in the H 1 - and L 2 -norms for elliptic problems and in the L ( H 1 )- and L ( L 2 )-norms for parabolic problems. Superconvergence of the difference between the gradients of the finite-volume solution and the interpolant can also be derived. Numerical results confirm the proved error estimates.

Description: A linear parabolic differential equation on a moving surface is discretized in space by evolving-surface finite elements and in time by backward difference formulas (BDFs). Using results from Dahlquist's G-stability theory and Nevanlinna & Odeh's multiplier technique together with properties of the spatial semidiscretization, stability of the full discretization is proved for BDF methods up to order 5 and optimal-order convergence is shown. Numerical experiments illustrate the behaviour of the fully discrete method.

Description: In this article, we develop the a priori and a posteriori error analysis of hp -version interior penalty discontinuous Galerkin finite element methods for strongly monotone quasi-Newtonian fluid flows in a bounded Lipschitz domain R d , d = 2, 3. In the latter case, computable upper and lower bounds on the error are derived in terms of a natural energy norm, which are explicit in the local mesh size and local polynomial degree of the approximating finite element method. A series of numerical experiments illustrate the performance of the proposed a posteriori error indicators within an automatic hp -adaptive refinement algorithm.

Description: In recent years, it has become increasingly clear that the critical issue in gradient methods is the choice of the step length, whereas using gradient as the search direction may lead to very effective algorithms, whose surprising behaviour has only been partially explained, mostly in terms of the spectrum of the Hessian matrix. On the other hand, the convergence of the classical Cauchy steepest descent (SD) method has been analysed extensively and related to the spectral properties of the Hessian matrix, but the connection with the spectrum of the Hessian has not been exploited much to modify the method in order to improve its behaviour. In this work, we show how, for convex quadratic problems, moving from some theoretical properties of the SD method, second-order information provided by the step length can be exploited to dramatically improve the usually poor practical behaviour of this method. This allows us to achieve computational results comparable with those of the Barzilai and Borwein algorithm, with the further advantage of monotonic behaviour.

Description: The so-called inductive McKay condition on finite simple groups, due to Isaacs–Malle–Navarro, has recently been reformulated by Späth. We show that this reformulation applies to the reduction theorem for Alperin's weight conjecture, due to Navarro–Tiep. This also simplifies the checking of the inductive condition for Alperin's weight conjecture in the case of simple groups of Lie type with regard to the defining prime.

Description: We construct a family of minimal smooth surfaces of general type with K 2 = 3 and p g = 0, which are finite (Z/2Z) 2 -covers of the 4-nodal cubic surface. This turns out to be a five-dimensional subfamily of the six-dimensional family constructed by Mendes Lopes and Pardini, which realizes the Keum–Naie surfaces with K 2 = 3 as degenerations. We show that the base of the Kuranishi family of a general surface in our subfamily is smooth. We prove that the closure of the corresponding subset of the Keum–Naie–Mendes Lopes–Pardini surfaces is an irreducible component of the Gieseker moduli space. As an important byproduct, it is shown that, for the surfaces in this irreducible component, the degree of the bicanonical map can only be 2 or 4.

Description: We show that the isometry group of the bounded Urysohn space is a simple group.

Description: In this paper we study Koszul cohomology and the Green and Prym–Green conjectures for canonical and Prym-canonical binary curves. We prove that, if property N p holds for a canonical or a Prym-canonical binary curve of genus g , then it holds for a generic canonical or Prym-canonical binary curve of genus g + 1. We also verify the Green and Prym–Green conjectures for generic canonical and Prym-canonical binary curves of low genus (6 ≤ g ≤ 15, g != 8 for Prym-canonical and 3 ≤ g ≤ 12 for canonical).

Description: We construct an infinite-dimensional Banach space of continuous functions C ( K ) such that every one-to-one operator T : C ( K )-〉 C ( K ) is onto.

Description: Magma dynamics and time scales during the VEI 5, 2000 bp eruption of El Misti volcano, southern Peru (EM2000BP) are investigated to address cyclic explosive activity at this hazardous volcano. The 1·4 km 3 of pumice falls and flows have abundant mingled pumice of high-K, calc-alkaline rhyolite and andesite composition. Phenocryst zoning and compositions reveal mutual exchange of plagioclase between the two magmas; amphibole in the rhyolite was derived from the andesite. Amphiboles in the andesite are predominantly unrimmed crystals whereas those in the rhyolite mostly exhibit reaction rims. Phase equilibria indicate that the andesite formed at ~900–950°C and 2–3 kbar pressure and was water-saturated with 5·1–6·0 wt % H 2 O, broadly similar to El Misti magmas overall. Amphibole, plagioclase, Ti-magnetite, and two pyroxenes were the crystallizing phases. A separate rhyolite magma existed higher in the crust at a temperature of 816 ± 30°C and ~5% H 2 O in which only plagioclase and Fe–Ti oxides were stable. The lack of cognate amphibole in the rhyolite despite H 2 O saturation requires that it staged above the stability limit of amphibole (〈100 MPa). Exchange reactions in amphibole (dominantly pargasitic) and trace element partitioning in plagioclase indicate that both andesite and rhyolite magmas were broadly constant in temperature and H 2 O content. These constraints suggest that the initially separate rhyolite and deeper andesite magmas interacted by an initial andesite recharge event that resulted in mingling and crystal exchange. A period of 50–60 days is required for amphibole introduced into the rhyolite to develop reaction rims owing to decompression. These rims are dominated by plagioclase, a consequence of the Al-rich nature of the amphibole. The lack of reaction rims on amphibole in the andesite implicates a second, more-forceful and voluminous eruption-triggering recharge event during which andesite rose rapidly from source to surface in ≤5 days at ascent rates of at least 0·023 m s –1 . Further decompression-driven crystallization is recorded in plagioclase rims and microlite growth that may have contributed to a rapid increase in viscosity leading to explosive eruption. This VEI 5 plinian eruption shares characteristics with other explosive events at El Misti on a time scale of 2000–4000 years, suggesting periodic recharge-driven explosive activity.

Description: We report structural evidence of ductile strain localization in mantle pyroxenite from the spinel to plagioclase websterite transition in the Ronda Peridotite (southern Spain). Mapping shows that, in this domain, small-scale shear zones occurring at the base of the lithospheric section are systematically located within thin pyroxenite layers, suggesting that the pyroxenite was locally weaker than the host peridotite. Strain localization is associated with a sudden decrease of grain size and increasing volume fractions of plagioclase and amphibole as a result of a spinel to plagioclase phase transformation reaction during decompression. This reaction also fostered hydrogen extraction (‘dehydroxylation’) from clinopyroxene producing effective fluid saturation that catalyzed the synkinematic net-transfer reaction. This reaction produced fine-grained olivine and plagioclase, allowing the onset of grain-size sensitive creep and further strain localization in these pyroxenite bands. The strain localization in the pyroxenites is thus explained by their more fertile composition, which allowed earlier onset of the phase transition reactions. Geothermobarometry undertaken on compositionally zoned constituent minerals suggests that this positive feedback between reactions and deformation is associated with cooling from at least 1000°C to 700°C and decompression from 1·0 to 0·5 GPa.

Description: In recent years, there has been an enormous interest in developing methods for the approximation of manifold-valued functions. In this paper, we focus on the manifold of symmetric positive-definite (SPD) matrices. We investigate the use of SPD-matrix means to adapt linear positive approximation methods to SPD-matrix-valued functions. Specifically, we adapt corner-cutting subdivision schemes and Bernstein operators. We present the concept of admissible matrix means and study the adapted approximation schemes based on them. Two important cases of admissible matrix means are treated in detail: the exp–log and the geometric matrix means. We derive special properties of the approximation schemes based on these means. The geometric mean is found to be superior in the sense of preserving more properties of the data, such as monotonicity and convexity. Furthermore, we give error bounds for the approximation of univariate SPD-matrix-valued functions by the adapted operators.

Description: We present a mass-preserving scheme for the stochastic nonlinear Schrödinger equation with multiplicative noise of Stratonovich type. It is a splitting scheme and we present an explicit formula for solving the sub-step related to the nonlinear part. The scheme is unconditionally stable in the L 2 norm. For the linear stochastic Schrödinger equation, we prove that the scheme has a strong convergence rate in time equal to 1, which is not common for stochastic partial differential equations with noise depending on space and time.

Description: A numerical scheme for the approximation of the elastic flow of inextensible curves is devised and convergence of approximations to exact solutions of the nonlinear time-dependent partial differential equation is proved. The nonlinear, pointwise constraint of local length preservation is linearized about a previous solution in each time step which leads to a sequence of linear saddle-point problems. The spatial discretization is based on piecewise Bézier curves and the resulting semiimplicit scheme is unconditionally stable and convergent.

Description: We study the coercivity properties and the norm dependence on the wave-number k of certain regularized combined field boundary integral operators that we recently introduced for the solution of two- and three-dimensional acoustic scattering problems with Neumann boundary conditions. We show that in the case of circular and spherical boundaries, our regularized combined field boundary integral operators are L 2 coercive for large enough values of the coupling parameter, and that the norms of these operators are bounded by constant multiples of the coupling parameter. We establish that the norms of the regularized combined field boundary integral operators grow modestly with the wave-number k for smooth boundaries and we provide numerical evidence that these operators are L 2 coercive for two-dimensional starlike boundaries. We present and analyse a fully discrete collocation (Nyström) method for the solution of two-dimensional acoustic scattering problems with Neumann boundary conditions based on regularized combined field integral equations. In particular, for analytic boundaries and boundary data, we establish pointwise superalgebraic convergence rates of the discrete solutions.

Description: The classical theory of Gaussian quadrature assumes a positive weight function. We will show that in some cases Gaussian rules can be constructed with respect to an oscillatory weight, yielding methods with complex quadrature nodes and positive weights. These rules are well suited to highly oscillatory integrals because they attain optimal asymptotic order. We show that, for the Fourier oscillator, this approach yields the numerical method of steepest descent, a method with optimal asymptotic order that has previously been proposed for this class of integrals. However, the approach readily extends to more general kernels, such as Bessel functions that appear as the kernel of the Hankel transform.

Description: We consider anisotropic Allen–Cahn equations with interfacial energy induced by an anisotropic surface energy density . Assuming that is positive, positively homogeneous of degree 1, strictly convex in tangential directions to the unit sphere and sufficiently smooth, we show the stability of various time discretizations. In particular, we consider a fully implicit and a linearized time discretization of the interfacial energy combined with implicit and semiimplicit time discretizations of the double-well potential. In the semiimplicit variant, concave terms are taken explicitly. The arising discrete spatial problems are solved by globally convergent truncated nonsmooth Newton multigrid methods. Numerical experiments show the accuracy of the different discretizations. We also illustrate that pinch-off under anisotropic mean curvature flow is no longer invariant under rotation of the initial configuration for a fixed orientation of the anisotropy.

Description: In this paper, we define a new finite element method for numerically approximating the solution of a partial differential equation in a bulk region coupled with a surface partial differential equation posed on the boundary of the bulk domain. The key idea is to take a polyhedral approximation of the bulk region consisting of a union of simplices, and to use piecewise polynomial boundary faces as an approximation of the surface. Two finite element spaces are defined, one in the bulk region and one on the surface, by taking the set of all continuous functions which are also piecewise polynomial on each bulk simplex or boundary face. We study this method in the context of a model elliptic problem; in particular, we look at well-posedness of the system using a variational formulation, derive perturbation estimates arising from domain approximation and apply these to find the optimal-order error estimates. A numerical experiment is described which demonstrates the order of convergence.

Description: As a model of more general contour integration problems we consider the numerical calculation of high-order derivatives of holomorphic functions using Cauchy's integral formula. Bornemann (2011, Accuracy and stability of computing high-order derivatives of analytic functions by Cauchy integrals. Found. Comput. Math. , 11 , 1–63) showed that the condition number of the Cauchy integral strongly depends on the chosen contour and solved the problem of minimizing the condition number for circular contours. In this paper, we minimize the condition number within the class of grid paths of step size h using Provan's algorithm for finding a shortest enclosing walk in weighted graphs embedded in the plane. Numerical examples show that optimal grid paths yield small condition numbers even in those cases where circular contours are known to be of limited use, such as for functions with branch-cut singularities.

Description: Anisotropic meshes are important for efficiently resolving incompressible flow problems that include boundary layer or corner singularity phenomena. Unfortunately, the stability of standard inf–sup stable mixed approximation methods is prone to degeneracy whenever the mesh aspect ratio becomes large. As an alternative, a stabilized mixed approximation method is considered here. Specifically, a robust a priori error estimate for the local jump stabilized Q 1 – P 0 approximation introduced by Kechkar & Silvester (1992, Analysis of locally stabilized mixed finite element methods for the Stokes problem. Math. Comp. , 58 , 1–10) is established for anisotropic meshes. Our numerical results demonstrate that the stabilized Q 1 – P 0 method is competitive with the nonconforming, nonparametric, rotated approximation method introduced by Rannacher & Turek (1992, Simple nonconforming quadrilateral Stokes element. Numer. Meth. Partial Differential Equations , 8 , 97–111).

Description: This work is about the numerical solution of the time-domain Maxwell's equations in dispersive propagation media by a discontinuous Galerkin time-domain method. The Debye model is used to describe the dispersive behaviour of the media. The resulting system of differential equations is solved using a centred-flux discontinuous Galerkin formulation for the discretization in space and a second-order leapfrog scheme for the integration in time. The numerical treatment of the dispersive model relies on an auxiliary differential equation approach similar to that which is adopted in the finite difference time-domain method. Stability estimates are derived through energy considerations and convergence is proved for both the semidiscrete and the fully discrete schemes.

Description: The parabolic singularly perturbed problem u xx ( x , t ) – x α u t ( x , t ) = f ( x , t ) is considered on the rectangular domain = (0,1) x (0, T ] with Dirichlet initial and boundary conditions. Here, is a small positive parameter and α is a positive constant. This problem is degenerate since the coefficient x α of u t vanishes along the side x = 0 of . Bounds on the derivatives of u are used to design a nonuniform mesh and a finite difference method on this mesh is constructed to solve the problem numerically. As the solution u is not in general uniformly bounded with respect to in the maximum norm, the convergence analysis of the numerical method requires the use of some unusual barrier functions and a special weighted discrete norm. Numerical examples are provided to support the theoretical results.

Description: We consider the numerical approximation of a general second-order semilinear parabolic stochastic partial differential equation driven by multiplicative and additive space–time noise. We examine convergence of exponential integrators for multiplicative and additive noise. We consider noise that is in a trace class and give a convergence proof in the root-mean-square L 2 norm. We discretize in space with the finite element method and in our implementation we examine both the finite element and the finite volume methods. We present results for a linear reaction–diffusion equation in two dimensions as well as a nonlinear example of a two-dimensional stochastic advection–diffusion–reaction equation motivated from realistic porous media flow.

Description: Sparse grids (Zenger, C. (1990) Sparse grids. Parallel Algorithms for Partial Differential Equations (W. Hackbusch ed.) Notes on Numerical Fluid Dynamics 31. Proceedings of the Sixth GAMM-Seminar; Bungartz, H.-J. & Griebel, M. (2004) Sparse grids. Acta Numer. , 13 , 1–123.) are tailored to the approximation of smooth high-dimensional functions. On a d -dimensional tensor product space, the number of grid points is N = O( h –1 |log h | d –1 ), where h is a mesh parameter. The so-called combination technique, based on hierarchical decomposition and extrapolation, requires specific multivariate error expansions of the discretization error on Cartesian grids to hold. We derive such error expansions for linear difference schemes through an error correction technique of semi-discretizations. We obtain overall error formulae of the type = O ( h p |log h | d –1 ) and analyse the convergence, with its dependence on dimension and smoothness, by examples of linear elliptic and parabolic problems, with numerical illustrations in up to eight dimensions.

Description: The numerical simulation of two-phase flow in a porous medium may lead, when using coupled finite volume schemes on structured grids, to the appearance of the so-called Grid Orientation Effect (GOE). We propose in this paper a procedure to eliminate this phenomenon, based on the use of new fluxes with a new stencil in the discrete version of the convection equation, without changing the discrete scheme for computing the pressure field. Numerical results show that the GOE does not significantly decrease with the size of the discretization using the initial scheme on the coupled problem, but that it is efficiently suppressed by the new procedure, even on coarse meshes. A mathematical study, based on a weak BV inequality using the new fluxes, confirms the convergence of the modified scheme in a particular case.

Description: We provide the first examples of words in the free group of rank 2 that are not proper powers and for which the corresponding word maps are non-surjective on an infinite family of finite non-abelian simple groups.

Description: We show that a finitely generated, residually finite group has the Haagerup property (Gromov's a-T-menability) if and only if one (or equivalently, all) of its box spaces admits a fibred coarse embedding into Hilbert space. In contrast, the box spaces of a finitely generated, residually finite hyperbolic group with property (T) do not admit a fibred coarse embedding into Hilbert space, but do admit a fibred coarse embedding into an p -space for some p 〉2.

Description: The late Miocene and younger mafic back-arc lavas in the southern Puna of the central Andean plateau have been attributed to the aftermath of crustal and mantle lithospheric delamination or foundering. In this paper, we analyze in more detail the nature of the back-arc mafic suite magmas, including the conditions of magma generation in the mantle and of magma evolution during ascent and ponding in the crust, using extensive compositional data for phenocryst minerals and olivine-hosted melt inclusions in combination with published and new whole-rock chemical and isotopic data. We estimate that the primary melts last equilibrated with an enriched mantle source at temperatures near 1375°C and pressures near 2 GPa, which is near the base of the seismically determined ~60 km thick crust. A mantle source geochemically enriched by continental material introduced through delamination and subducted erosion processes is required to explain the coincidence of the high 87 Sr/ 86 Sr ratios (〉0·705) and high Sr concentrations (〉700 ppm) of the most primitive lavas (e.g. 9–10 wt % MgO, olivine Fo 88 ). The crystallization conditions inferred from mineral–melt equilibria indicate that olivine ( T = 1320–1220°C) was followed by clinopyroxene ( T = 1230–1140°C). Clinopyroxene–melt equilibration pressures of 0·7 to near 1 GPa in the most mafic samples indicate that the magmas crystallized at mid-crustal depths of 20–35 km, within a region of inferred partial melt accumulation based on the presence of low seismic velocity zones. Olivine-hosted melt inclusions indicate relatively dry melts (maximum 0·5 wt % H 2 O) with unusual high-Al basaltic compositions, which are attributed to the high-pressure suppression of plagioclase crystallization. A first stage of crustal contamination before mid-crustal accumulation and crystallization of the mafic magmas is suggested by high O-isotope ratios in olivine phenocrysts and negative Eu anomalies in clinopyroxene from the plagioclase-free mafic lavas. Mixing models based on trace elements and radiogenic isotopes suggest assimilation of silicic melt in the lower crust, similar to contemporaneous glassy dacites with steep REE patterns and negative Eu anomalies. A second stage of crustal assimilation at shallower depths is indicated by the mismatch of incompatible elements in clinopyroxene relative to bulk-rock compositions, by strong positive correlations of radiogenic isotopes with wt % SiO 2 , and by petrographic observation of partly resorbed and reacted quartz xenocrysts. Mixing calculations require the erupted magmas to have assimilated in total some 15–25% crust.

Description: Hornblende-bearing basanites and alkali basalts from the Rhön area of Germany (part of the Central European Volcanic Province; CEVP) have high TiO 2 (3–4 wt %), moderately high Mg# (mostly 〉0·50), variable Cr (400–30 ppm) and Ni (160–20 ppm) abundances, and are enriched in incompatible trace elements and rare earth elements (REE). In primitive mantle-normalized multi-element diagrams they show a strong depletion in Ba, Rb, and K relative to trace elements of similar incompatibility. Some alkali basalts and more differentiated rocks have lower Mg# and lower abundances of Ni and Cr, and have undergone fractionation of olivine, clinopyroxene, Fe–Ti oxides and amphibole. The trace element constraints (e.g. low Nb/U and Ce/Pb and the Nd–Sr–Pb isotope compositions of some basalts) indicate that assimilation of lower crustal material has modified the composition of the primary mantle-derived magmas. Most of the basanites and alkali basalts approach the Sr–Nd–Pb isotope compositions inferred for the EAR (European Asthenospheric Reservoir) component. Variations in REE abundances and correlations between REE ratios suggest partial melting of amphibole-bearing spinel peridotite containing a significant portion of non-peridotitic material (i.e. pyroxenite). The presence of residual amphibole, indicated by depletion of K and Rb relative to Ba and Nb, requires melting close to the asthenosphere–lithosphere boundary or within the lithospheric mantle, most probably of a veined mantle source. Temperature and pressure estimates indicate a depth of melting for the most primitive lavas at ~80 km at temperatures of ~1290°C. Based on Sr–Nd isotope and trace element constraints it is proposed that asthenospheric melts similar in composition to EAR melts observed elsewhere in the CEVP froze at the asthenosphere–lithosphere thermal boundary as veins in the lithospheric mantle. These veins were remelted after only short storage times by ascending asthenospheric melts, imposing the prominent amphibole signature upon the basalts. The fairly radiogenic Pb isotope signatures are expected to originate from melting of enriched, low melting temperature components incorporated in the depleted upper (asthenospheric) mantle and therefore do not require upwelling of deep-seated mantle sources for the Rhön or many other continental alkaline lavas with similar Pb isotope signatures.

Description: Two mafic eruptive products from Vesuvius, a tephrite and a trachybasalt, have been crystallized in the laboratory to constrain the nature of primitive Vesuvius magmas and their crustal evolution. Experiments were performed at high temperatures (from 1000 to ≥1200°C) and both at 0·1 MPa and at high pressures (from 50 to 200 MPa) under H 2 O-bearing fluid-absent and H 2 O- and CO 2 -bearing fluid-present conditions. Experiments started from glass except for a few that started from glass plus San Carlos olivine crystals to force olivine saturation. Melt H 2 O concentrations reached a maximum of 6·0 wt % and experimental f O 2 ranged from NNO – 0·1 to NNO + 3·4 (where NNO is nickel–nickel oxide buffer). Clinopyroxene (Mg# up to 93) is the liquidus phase for the two investigated samples; it is followed by leucite for H 2 O in melt 〈3 wt %, and by phlogopite (Mg# up to 81) for H 2 O in melt 〉3 wt %. Olivine (Fo 85 ) crystallized spontaneously in only one experimental charge. Plagioclase was not found. Upon progressive crystallization of clinopyroxene, glass K 2 O and Al 2 O 3 contents strongly increase whereas MgO, CaO and CaO/Al 2 O 3 decrease; the residual melts follow the evolution of Vesuvius whole-rocks from trachybasalt to tephrite, phonotephrite and to tephriphonolite. Concentrations of H 2 O and CO 2 in near-liquidus 200 MPa glasses and primitive melt inclusions from the literature overlap. The earliest evolutionary stage, corresponding to the crystallization of Fo-rich olivine, was reconstructed by the olivine-added experiments. They show that the primitive Vesuvius melts are trachybasalts (K 2 O ~ 4·5–5·5 wt %, MgO = 8–9 wt %, Mg# = 75–80, CaO/Al 2 O 3 = 0·9–0·95) that crystallize Fo-rich olivine (90–91) as the liquidus phase between 1150 and 1200°C and from 300 to 〈200 MPa. Primitive Vesuvius melts are volatile-rich (1·5–4·5 wt % H 2 O and 600–4500 ppm CO 2 in primitive melt inclusions) and oxidized (from NNO + 0·4 to NNO + 1·2). Assimilation of carbonate wall-rocks by ascending primitive magmas can account for the disappearance of olivine from crystallization sequences and explains the lack of rocks representative of olivine-crystallizing magmas. A correlation between carbonate assimilation and the type of feeding system is proposed: carbonate assimilation is promoted for primitive magma batches of small volumes. In contrast, for longer-lived, large-volume, less frequently recharged, hence more evolved, cooler reservoirs, magma–carbonate interaction is limited. Primitive magmas from Vesuvius and other Campanian volcanoes have similar redox states. However, the Cr# of Vesuvius spinels is distinctive and therefore the peridotitic component in the mantle source of Vesuvius differs from that of the other Campanian magmas.

Description: Peridotite xenoliths exhumed by Quaternary alkaline magmatism in the Tahalgha district, southern Hoggar, represent fragments of the subcontinental lithospheric mantle beneath the boundary between the two major structural domains of the Tuareg Shield: the ‘Polycyclic Central Hoggar’ to the east and the ‘Western Hoggar’, or ‘Pharusian Belt’, to the west. Samples were collected from volcanic centres located on both sides of a major lithospheric shear zone at 4°35' separating these two domains. Although showing substantial variations in their deformation microstructures, equilibrium temperatures and modal and chemical compositions, the studied samples do not display any systematic changes of these features across the 4°35' fault. The observed variations rather record small-scale heterogeneities distributed throughout the study area and reflecting the widespread occurrence of vein conduits and metasomatized wall-rocks related to trans-lithospheric melt circulation during the Cenozoic. These features include partial annealing of pre-existing deformation microstructures, post-deformation metasomatic reactions, and trace-element enrichment, coupled with heating from 750–900°C (low-temperature lherzolites) to 900–1150°C (intermediate- T lherzolites and high- T harzburgites and wehrlites). Trace-element modelling confirms that the range of rare earth element (REE) variations observed in the Tahalgha clinopyroxenes may be accounted for by reactive porous flow involving a single stage of basaltic melt infiltration into a light REE (LREE)-depleted protolith. Whole-rock compositions record the final entrapment of disequilibrium metasomatic melts upon thermal relaxation of the veins–wall-rock system. The striking correlations between equilibrium temperatures and trace-element enrichment favor a scenario in which the high-temperature peridotites record advective heat transport along melt conduits, whereas the intermediate- and low-temperature lherzolites reflect conductive heating of the host Mechanical Boundary Layer. This indicates that the lithosphere did not reach thermal equilibrium, suggesting that the inferred heating event was transient and was rapidly erased by thermal relaxation down to the relatively low-temperature present-day geotherm. The low- T (〈900°C) deformed lherzolites (porphyroclastic to equigranular) are characterized by only incipient annealing and LREE-depleted clinopyroxene compositions. They were only weakly affected by the Cenozoic events and could represent relatively well-preserved samples from rejuvenated Pan-African lithosphere. Extensive lithospheric rejuvenation occurred either regionally during the Pan-African orogeny, as a result of lithospheric delamination or thermomechanical erosion after thickening, or more locally along the meridional shear zones. The low- T Tahalgha lherzolites are comparable with lherzolites from Etang de Lherz, southern France, interpreted as lithospheric mantle rejuvenated by melt-induced refertilization during a late stage of the Variscan orogeny.

Description: Hydrothermal experiments were conducted at 200 MPa and 900–1018°C to determine the solubilities, fluid(s)–melt partitioning, and mixing properties of H 2 O, CO 2 , S, Cl, and F in phonolitic–trachytic melts saturated in vapor, vapor plus saline liquid, or saline liquid. The bulk compositions and S, Cl, and F concentrations of the run-product glasses were determined by electron microprobe and the H 2 O and CO 2 contents by Fourier-transform infrared spectroscopy (FTIR). A new parameterization was developed to calculate molar absorption coefficients for FTIR analysis of carbonate in glasses and applied to the run-product glasses. The concentrations of volatiles in the fluid(s) were determined by mass-balance calculations and checked with chloridometer analysis and gravimetry. The range in oxygen fugacity of these experiments is NNO to NNO + 2 (where NNO is nickel–nickel oxide buffer). The phonolitic–trachytic melts dissolved up to 7·5 wt % H 2 O, 0·94 wt % Cl, 0·73 wt % CO 2 , 0·75 wt % F, and 0·16 wt % S, and the integrated bulk fluid(s) contained up to 99 mol % H 2 O, 34 mol % Cl, 82 mol % CO 2 , 1·7 mol % F, and 3·7 mol % S. The mixing relationships of H 2 O, CO 2 , and Cl in melt versus fluid(s) are complex and strongly non-ideal at these pressure–temperature conditions, particularly with two fluid phases stable. The concentrations of H 2 O and CO 2 in melt change with the addition of Cl ± S to the system, and the solubility of Cl in melt varies with S. The reductions in H 2 O and CO 2 solubility in melt exceed those resulting from simple dilution of the coexisting fluid(s) owing to addition of other volatiles. The partitioning of H 2 O and CO 2 between fluid(s) and melt varies as a function of fluid(s) and melt composition. The experimental data are applied to phonolitic and related magmas of Mt. Somma–Vesuvius, Italy, Mt. Erebus, Antarctica, and Cripple Creek, USA, to better interpret processes of fluid(s) exsolution in eruptive and mineralizing systems. Application of the experimental results also provides constraints on eruptive and mineralizing fluid(s) compositions.

Description: The early stages of magmatic processes operating at mantle depths beneath continental arcs are poorly known. The chemical compositions of minerals and rocks, mineral Sr–Nd–Hf–O isotopes and zircon U–Pb ages of garnet clinopyroxenite dykes from the Shenglikou peridotite massif (North Qaidam Orogen, NE Tibet, China) were studied to constrain their sources and genesis, and the dynamic processes that controlled pyroxenite formation beneath an early Paleozoic active continental margin. Major-element compositions of bulkrocks suggest that the pyroxenitic protoliths were cumulates segregated from a melt, which was extracted from a peridotite-dominated mantle source. Bulk-rock and mineral trace-element patterns show strong enrichment in fluid-mobile elements (e.g. Cs, Rb, Ba, Th, U, K, Pb and Li) and marked negative anomalies in the high field strength elements relative to rare earth elements, similar to the characteristics of melts derived from a volatile-rich sub-arc mantle. Enriched Sr and Nd initial isotopic compositions at 500 Ma ( 87 Sr/ 86 Sr of 0·70919–0·71774 and Nd of –16·3 to –3·4) are in contrast to the highly radiogenic Hf isotope compositions (similar to those of the depleted-mantle reservoir) and to the uncontaminated upper-mantle 18 O V-SMOW (garnet: 5·6 ± 0·3, 2SD, n = 61; zircon: 5·9 ± 0·3, 2SD, n = 28). These decoupled isotopic signatures suggest that the melt source was located in a convective mantle wedge (controlling the Hf and O isotopes) that had been pervasively metasomatized by fluids from a subducted Proto-Tethys oceanic slab (controlling the Sr–Nd isotopes and highly incompatible elements). Zircons with two groups of U–Pb ages (430 ± 5 Ma and 401 ± 7 Ma) were generated by recrystallization events, corresponding to UHP metamorphism and a major uplift stage during the North Qaidam orogeny, respectively. The combined evidence reveals a picture of continental arc magmatism at mantle depths and subsequent continental collision. The subduction of the Proto-Tethys oceanic slab beneath the southern Qilian margin triggered flux melting of the metasomatized convective mantle wedge and generated hydrous arc magmas. These primitive magmas intruded into the overlying lithospheric mantle and segregated the cumulates parental to the Shenglikou pyroxenites. Subsequent continental subduction incorporated fragments of the mantle-wedge peridotite (containing pyroxenite dykes) at ~430 Ma and carried them to shallow depths during exhumation at ~400 Ma.

Description: Using a new high-resolution dataset, this study presents evidence for short length scale 18 O/ 16 O heterogeneity in the mantle source region of young (age 12 ka bp ) Icelandic basalts. The dataset comprises secondary ion mass spectrometry determinations of 18 O/ 16 O in single compositional zones of plagioclase crystals from the primitive Borgarhraun flow in northern Iceland, along with trace and major element data from the same zones. The presence of mantle under Iceland with 18 O below typical mid-ocean ridge basalt (MORB) values of ~5·5 ± 0·3 (VSMOW) has previously been disputed, because variability in 18 O in many Icelandic basalts is also known to be caused by the interaction of basaltic melts with crustal lithologies that have been altered by low- 18 O meteoric water. Primitive basalt flows, such as Borgarhraun, and their macrocrysts are the most likely candidates to retain a mantle 18 O signature. However, the role of crustal processes in generating the low 18 O in olivine crystals from these flows has not unequivocally been ruled out. By making intra-crystal analyses in Borgarhraun plagioclase it has been possible in this study to obtain a detailed record of the chemical and isotopic compositions of the melts that crystallized the plagioclase zones. The variability observed in trace element compositions of the early crystallized anorthitic plagioclase zones (80·9–89·4 mol % anorthite) is firstly shown to arise from melt compositional variability, and equilibrium melt concentrations of Sr, La and Y are then calculated from the crystal concentrations of these elements using carefully selected partition coefficients. The ranges of incompatible trace element ratios (La/Y, Sr/Y) in these equilibrium melts reflect a range of compositions of fractional mantle melts, a result that is in agreement with previous proposals for the cause of variability in trace element indices of Borgarhraun olivine-hosted melt inclusions and clinopyroxene compositional zones. Correlations observed between La/Y and Sr/Y in the melts in equilibrium with the Borgarhraun plagioclase zones and the 18 O of these zones therefore support the hypothesis that the mantle under Iceland is heterogeneous in 18 O/ 16 O. Such correlations have not previously been observed in intra-crystal data from Iceland, and provide strong evidence that mantle material with abnormally low 18 O may exist in the form of readily fusible heterogeneities alongside ambient mantle with MORB-like 18 O (+5·5) on a length scale of 〈100 km. The lowest 18 O of plagioclase that is attributed to a mantle origin in this study is 4·5 ± 0·4, equating to a melt equivalent value of 4·3 ± 0·5 or an olivine equivalent value of 3·8 ± 0·5.

Description: Mafic to ultramafic intrusions of the Qullinaaraaluk suite (Q-suite) were emplaced into the Ungava craton of the Northeastern Superior Province during an episode of intense igneous activity and crustal reworking from c. 2·74 to 2·70 Ga. Orthopyroxene-rich Q-suite intrusions from the Hudson Bay Terrane and southwestern Rivière Arnaud Terrane, and orthopyroxene-poor Q-suite intrusions from the north–central Rivière Arnaud Terrane indicate the existence of at least two Q-suite magma types: a subalkaline magma parental to the orthopyroxene-rich intrusions and a transitional magma parental to the orthopyroxene-poor intrusions. Both types of intrusions are characterized by light rare earth element (LREE)-enriched, high field strength element (HFSE)-depleted trace element profiles that reflect, in large part, contamination by the tonalite–trondhjemite–granodiorite-dominated crust. Near-chondritic to strongly sub-chondritic initial Nd (2·72 Ga) values (+2 to –10) of the Q-suite intrusions reflect the combined effects of both the amount of crustal contamination and the age-dependent isotopic composition of the contaminant. The inferred trace element profiles of the uncontaminated Q-suite magmas were probably flat to LREE-depleted. The transitional magmas that produced the least evolved dunitic cumulates of the Q-suite were ferropicrites (MgO ~14 wt %, FeO TOT ~17 wt %). In contrast, the magmas parental to the primitive Q-suite harzburgites were Fe-rich, high-Mg basalts (MgO ~11 wt %; FeO ~14 wt %). The high Fe contents of the Q-suite magmas are incompatible with derivation from a pyrolitic mantle [Mg-number ~0·90, Mg/(Mg + Fe TOT )] and require sources significantly enriched in iron (Mg-number ≤0·79). Both magma types are also characterized by relatively low Ni contents suggesting derivation from source regions depleted in Ni relative to pyrolitic mantle peridotite. Differences in the major element compositions of the subalkaline and transitional parental magmas may reflect compositional diversity among the Fe-rich mantle sources. Comparisons with melting experiments on compositions analogous to the Martian mantle suggest that the Q-suite magmas may rather be generated by different degrees of melting of a common source with an Fe content slightly lower than that of the Homestead L5 ordinary chondrite (Mg-number = 0·77). The Fe-rich picritic to high-Mg basaltic magmas last equilibrated with garnet-free harzburgitic to lherzolitic residues at upper mantle pressures (≤5 GPa). The craton-wide occurrence of c. 2·72–2·70 Ga Q-suite mafic to ultramafic plutons suggests that underplating by Fe-rich mantle melts may have had a key role in the c. 2·74–2·70 Ga cratonization of the Northeastern Superior Province.

Description: Arc basalts are more oxidized than mid-ocean ridge basalts, but it is unclear whether this difference is due to differentiation processes in the Earth’s crust or to a fundamental difference in the oxygen fugacity of their mantle sources. Distinguishing between these two hypotheses is important for understanding redox-sensitive processes related to arc magmatism, and thus more broadly how Earth materials cycle globally. We present major, volatile, and trace element concentrations in combination with Fe 3+ /Fe ratios determined in olivine-hosted glass inclusions and submarine glasses from five Mariana arc volcanoes and two regions of the Mariana Trough. For single eruptions, Fe 3+ /Fe ratios vary along liquid lines of descent that are either slightly oxidizing (olivine + clinopyroxene + plagioclase fractionation, CO 2 ± H 2 O degassing) or reducing (olivine + clinopyroxene + plagioclase ± magnetite fractionation, CO 2 + H 2 O + S degassing). Mariana samples are consistent with a global relationship between calc-alkaline affinity and both magmatic H 2 O and magmatic oxygen fugacity, where wetter, higher oxygen fugacity magmas display greater affinity for calc-alkaline differentiation. We find, however, that low-pressure differentiation cannot explain the majority of variations observed in Fe 3+ /Fe ratios for Mariana arc basalts, requiring primary differences in magmatic oxygen fugacity. Calculated oxygen fugacities of primary mantle melts at the pressures and temperatures of melt segregation are significantly oxidized relative to mid-ocean ridge basalts (~QFM, where QFM is quartz–fayalite–magnetite buffer), ranging from QFM + 1·0 to QFM + 1·6 for Mariana arc basalts, whereas back-arc related samples record primary oxygen fugacities that range from QFM + 0·1 to QFM + 0·5. This Mariana arc sample suite includes a diversity of subduction influences, from lesser influence of a homogeneous H 2 O-rich component in the back-arc, to sediment melt- and fluid-dominated influences along the arc. Primary melt oxygen fugacity does not correlate significantly with sediment melt contributions (e.g. Th/La), nor can it be attributed to previous melt extraction in the back-arc. Primary melt oxygen fugacity correlates strongly with indices of slab fluids (e.g. Ba/La) from the Mariana Trough through the Mariana arc, increasing by 1·5 orders of magnitude as Ba/La increases by a factor of 10 relative to mid-ocean ridge basalts. These results suggest that contributions from the slab to the mantle wedge may be responsible for the elevated oxygen fugacity recorded by Mariana arc basalts and that slab fluids are potentially very oxidized.

Description: Magma mixing and crystal mush disaggregation are important processes in basaltic magma reservoirs. We carried out a detailed petrological and geochemical study on a highly plagioclase-phyric eruption within the Eastern Volcanic Zone of Iceland—the Skuggafjöll eruption—to investigate crystal storage and transport processes within a single magmatic system. Crystal content and phase proportions vary between samples: the least phyric samples have phase proportions similar to the low-pressure, three-phase gabbro eutectic (plg:cpx:ol ~ 11:6:3), whereas highly phyric samples are strongly enriched in plagioclase (plg:cpx:ol ~ 8:1:1). Statistically significant geochemical variability in 28 whole-rock samples collected across the eruption can be accounted for by variable accumulation of a troctolitic assemblage containing plagioclase and olivine in an approximately 9:1 ratio. Two macrocryst assemblages are defined using compositional and textural information recorded in QEMSCAN® images: a primitive assemblage of high-anorthite plagioclase (An 〉83 ) and high-forsterite olivine (Fo 〉84 ), and an evolved assemblage of low-anorthite plagioclase (An 〈79 ), low-forsterite olivine (Fo 〈82 ) and clinopyroxene (Mg# ~ 82). Plagioclase and olivine have strongly bimodal composition distributions whereas the composition distribution of clinopyroxene is unimodal. The mean trace element composition of melt inclusions hosted within high-forsterite olivine and high-anorthite plagioclase macrocrysts is the same (mean Ce/Y ~ 0·47–0·48), confirming that both primitive macrocryst phases crystallized from the same distribution of melts. Clinopyroxene macrocrysts and matrix glasses are in Ce/Yb equilibrium with each other, indicating that the evolved assemblage crystallized from melts with a more incompatible trace element-enriched composition (mean Ce/Y ~ 0·65–71) than the primitive assemblage. Variability in whole-rock, macrocryst and melt inclusion compositions suggests that the Skuggafjöll magma experienced two stages of crystallization. Primitive macrocrysts crystallized first from incompatible trace element-depleted melts within a shallow crustal magma reservoir. These primitive macrocrysts were subsequently stored in crystal mushes that ultimately disaggregated into an evolved and incompatible trace element-enriched magma from which the evolved assemblage crystallized. On average, ~17% of the erupted magma at Skuggafjöll is composed of accumulated macrocrysts entrained from crystal mushes. The timescale between mush disaggregation and eruption, during which crystal accumulation occurred, was short—of the order of years—according to simple diffusion calculations. Striking petrological similarities between Skuggafjöll and other highly phyric eruptions both in Iceland and along mid-ocean ridges indicate that crystal accumulation by mush disaggregation is likely to be an important mechanism for generating highly phyric magmas in basaltic plumbing systems.

Description: We present experimental data on the partitioning of Li, Be, B, K, Mg, Sr, Ga, Rb, Cs, Ba, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu, Th, U, Hf, Zr, Nb and Ta between lawsonite and fluid, and zoisite and fluid at 3·0–3·5 GPa and 650–850°C. The aim is to provide data bearing on the trace element contents of fluids released during dehydration of subducting oceanic crust. Experimental trace element partition coefficients for lawsonite indicate a preference for the light rare earth elements (LREE) over the heavy REE (HREE) and for Be. These characteristics are consistent with the chemical composition of lawsonite in natural rocks. Experimental trace element partition coefficients for zoisite indicate a preference for HREE relative to LREE. This observation, consistent with earlier experimental data, is the reverse of the observed trace element compositions of natural zoisites, indicating the influence of other factors on the trace element contents of this phase. Lattice strain theory explains well the experimentally derived partitioning of divalent cations in the Ca-site between lawsonite and fluid. However, the weak relative fractionation of REE between lawsonite and fluid cannot be explained by lattice strain theory, as previously observed for zoisite–fluid REE partitioning. We combine our experimental data with thermodynamic models of mineral stability to model the compositions of fluids released during subduction of altered normal mid-ocean ridge basalt. The low La/Sm ratio associated with very high Ba/Th in arc magmas can be explained only if allanite is stable in the subducting oceanic crust. This suggests that the crustal fluid component involved in arc magma petrogenesis results from processes occurring in the warm, top part of the subducting slab. Decreasing lawsonite modal proportion with depth is associated with a large release of fluid characterized by low B/Be ratios that could explain the decreasing B/Be ratios in arc magmas with increasing distance from the trench. This implies that an important Be input in arc magma originates from the fluid generated during oceanic crust dehydration.

Description: To contribute to our understanding of the mechanisms and pathways of fluid movement through deeply subducted crust, we investigate high-pressure veins cutting eclogite-facies (~2·0 GPa and ~600°C) metagabbros of the Monviso Ophiolite, Italian Western Alps. The veins consist mainly of omphacite with minor garnet, rutile, talc and accessory zircon. Most of the vein minerals have major and trace element compositions that are comparable with the host-rock minerals, and vein and host-rock zircons have similar Hf isotopic compositions. These observations support the conclusions of previous studies that these veins largely formed from a locally sourced hydrous fluid during prograde or peak metamorphism. However, the bulk-rock Cr and Ni contents of the veins are significantly higher than those of the surrounding host eclogites. We also document distinct Cr-rich (up to weight per cent levels) zones in omphacite, garnet and rutile in some vein samples. Vein garnet and talc also have relatively high MgO and Ni contents. X-ray maps of vein garnet and rutile grains reveal complex internal zoning features, which are largely defined by micrometre-scale variations in Cr content. Some grains have concentric and oscillatory zoning in Cr, whereas others feature a chaotic fracture-like pattern. These Cr-rich zones are associated with high concentrations of Ni, B, As, Sb, Nb, Zr and high ratios of light rare earth elements (LREE) to middle REE (MREE) compared with low-Cr vein and host-rock minerals. Petrological and mass-balance constraints verify that the Cr-rich zones in the veins were not derived from internally sourced fluids, but represent precipitates from an external fluid. The external source that is consistent with the distinctive trace element characteristics of the vein components is antigorite serpentinite, which forms the structural basement of the high-pressure metagabbros. We propose at least two separate growth mechanisms for the Monviso veins. Most vein infillings were formed during progressive prograde metamorphism from locally derived fluid. Influx of the serpentinite-derived or other external fluid was transient and episodic and was probably achieved via brittle fractures, which preferentially formed along the pre-existing vein structures. The dehydration of serpentinite at high pressures in subduction zones may provide crucial volatiles and trace elements for arc magmas. Our results indicate that the movement of these fluids through subducted oceanic crust is likely to be highly channeled and transient so the progressive development of vein systems in mafic rocks may also be crucial for forming channelways for long-distance fluid flow at depth in subduction zones.

Description: Siwi caldera, in the Vanuatu arc (Tanna island), is a rare volcanic complex where both persistent eruptive activity (Yasur volcano) and rapid block resurgence (Yenkahe horst) can be investigated simultaneously during a post-caldera stage. Here we provide new constraints on the feeding system of this volcanic complex, based on a detailed study of the petrology, geochemistry and volatile content of Yasur–Siwi bulk-rocks and melt inclusions, combined with measurements of the chemical composition and mass fluxes of Yasur volcanic gases. Major and trace element analyses of Yasur–Siwi volcanic rocks, together with literature data for other volcanic centers, point to a single magmatic series and possibly long-lived feeding of Tanna volcanism by a homogeneous arc basalt. Olivine-hosted melt inclusions show that the parental basaltic magma, which produces basaltic-trachyandesites to trachyandesites by ~50–70% crystal fractionation, is moderately enriched in volatiles (~1 wt % H 2 O, 0·1 wt % S and 0·055 wt % Cl). The basaltic-trachyandesite magma, emplaced at between 4–5 km depth and the surface, preserves a high temperature (1107 ± 15°C) and constant H 2 O content (~1 wt %) until very shallow depths, where it degasses extensively and crystallizes. These conditions, maintained over the past 1400 years of Yasur activity, require early water loss during basalt differentiation, prevalent open-system degassing, and a relatively high heat flow (~10 9 W). Yasur volcano releases on average ≥ 13·4 x 10 3 tons d –1 of H 2 O and 680 tons d –1 of SO 2 , but moderate amounts of CO 2 (840 tons d –1 ), HCl (165 tons d –1 ), and HF (23 tons d –1 ). Combined with melt inclusion data, these gas outputs constrain a bulk magma degassing rate of ~5 x 10 7 m 3 a –1 , about a half of which is due to degassing of the basaltic-trachyandesite. We compute that 25 km 3 of this magma have degassed without erupting and have accumulated beneath Siwi caldera over the past 1000 years, which is one order of magnitude larger than the accumulated volume uplift of the Yenkahe resurgent block. Hence, basalt supply and gradual storage of unerupted degassed basaltic-trachyandesite could easily account for (or contribute to) the Yenkahe block resurgence.

Description: Primitive basalts are rarely found in arcs. The active NW Rota-1 volcano in the Mariana arc has erupted near-primitive lavas, which we have sampled with ROV Hyper-Dolphin (HPD). Samples from the summit (HPD480) and eastern flank (HPD488) include 17 magnesian basalts (51–52 wt % SiO 2 ) with 7·5–9·5 wt % MgO and Mg-number of 61–67, indicating little fractionation. Olivine phenocrysts are as magnesian as Fo 93 and contain 0·4 wt % NiO; the Cr/(Cr + Al) values of spinels are mostly 0·5–0·8, indicating equilibrium with depleted mantle. There are three petrographic groups, based on phenocryst populations: (1) cpx–olivine basalt (COB); (2) plagioclase–olivine basalt (POB); (3) porphyritic basalt. Zr/Y and Nb/Yb are higher in POB (3·1–3·2 and 1·2–1·5, respectively) than in COB (Zr/Y = 2·8–3·0 and Nb/Yb = 0·7–0·9), suggesting that POB formed from lower degrees of mantle melting, or that the COB mantle source was more depleted. On the other hand, COB have Ba/Nb (70–80) and Th/Nb (0·4–0·5) that are higher than for POB (Ba/Nb = 30–35 and Th/Nb = 0·1–0·2), and also have steeper light rare earth element (LREE)-enriched patterns. Moreover, COB have enriched 87 Sr/ 86 Sr and 143 Nd/ 144 Nd, and higher Pb isotope values, suggesting that COB has a greater subduction component than POB. 176 Hf/ 177 Hf between COB and POB are similar and Hf behavior in COB and POB is similar to that of Zr, Y and HREE, suggesting that Hf is not included in the subduction component, which produced the differences between COB and POB. The calculated primary basaltic magmas of NW Rota-1 volcano (primary COB and POB magmas) indicate segregation pressures of 2–1·5 GPa (equivalent to 65–50 km depth). These magmas formed by 24–18% melting of mantle peridotite having Mg-number ~89·5. Diapiric ascent of hydrous peridotite mixed heterogeneously with sediment melts may be responsible for the NW Rota-1 basalts. These two basalt magma types are similar to those found at Sumisu and Torishima volcanoes in the Izu–Bonin arc, with COB representing wetter and POB representing drier magmas, where subduction zone-derived melt components are coupled with the water contents.

Description: The origin of andesite is an important issue in petrology because andesite is the main eruptive product at convergent margins, corresponds to the average crustal composition and is often associated with major Cu–Au mineralization. In this study we present petrographic, mineralogical, geochemical and isotopic data for basaltic andesites of the latest Pleistocene Pilavo volcano, one of the most frontal volcanoes of the Ecuadorian Quaternary arc, situated upon thick (30–50 km) mafic crust composed of accreted Cretaceous oceanic plateau rocks and overlying mafic to intermediate Late Cretaceous–Late Tertiary magmatic arcs. The Pilavo rocks are basaltic andesites (54–57·5 wt % SiO 2 ) with a tholeiitic affinity as opposed to the typical calc-alkaline high-silica andesites and dacites (SiO 2 59–66 wt %) of other frontal arc volcanoes of Ecuador (e.g. Pichincha, Pululahua). They have much higher incompatible element contents (e.g. Sr 650–1350 ppm, Ba 650–1800 ppm, Zr 100–225 ppm, Th 5–25 ppm, La 15–65 ppm) and Th/La ratios (0·28–0·36) than Pichincha and Pululahua, and more primitive Sr ( 87 Sr/ 86 Sr ~0·7038–0·7039) and Nd ( Nd ~ +5·5 to +6·1) isotopic signatures. Pilavo andesites have geochemical affinities with modern and recent high-MgO andesites (e.g. low-silica adakites, Setouchi sanukites) and, especially, with Archean sanukitoids, for both of which incompatible element enrichments are believed to result from interactions of slab melts with peridotitic mantle. Petrographic, mineral chemistry, bulk-rock geochemical and isotopic data indicate that the Pilavo magmatic rocks have evolved through three main stages: (1) generation of a basaltic magma in the mantle wedge region by flux melting induced by slab-derived fluids (aqueous, supercritical or melts); (2) high-pressure differentiation of the basaltic melt (at the mantle–crust boundary or at lower crustal levels) through sustained fractionation of olivine and clinopyroxene, leading to hydrous, high-alumina basaltic andesite melts with a tholeiitic affinity, enriched in incompatible elements and strongly impoverished in Ni and Cr; (3) establishment of one or more mid-crustal magma storage reservoirs in which the magmas evolved through dominant amphibole and clinopyroxene (but no plagioclase) fractionation accompanied by assimilation of the modified plutonic roots of the arc and recharge by incoming batches of more primitive magma from depth. The latter process has resulted in strongly increasing incompatible element concentrations in the Pilavo basaltic andesites, coupled with slightly increasing crustal isotopic signatures and a shift towards a more calc-alkaline affinity. Our data show that, although ultimately originating from the slab, incompatible element abundances in arc andesites with primitive isotopic signatures can be significantly enhanced by intra-crustal processes within a thick juvenile mafic crust, thus providing an additional process for the generation of enriched andesites.

Description: Mid-ocean ridge basalts (MORB) from the Arctic Ocean have been significantly less studied than those from other oceans. The Arctic ridges (Gakkel Ridge and Lena Trough) are ultraslow-spreading ridges with low melt productivity and are thus the best locations to investigate mantle heterogeneity. We report the major and trace element and Sr–Nd–Pb–Hf isotope compositions of basalts generated along the Lena Trough and the westernmost part of the Gakkel Ridge in the Arctic Ocean. Basalts from the northern Lena Trough and westernmost Gakkel Ridge (NLT–WGR) have compositions close to normal MORB. The geochemical composition of the NLT–WGR lavas confirms a binary mixing model involving melts from a depleted MORB mantle source and a Spitsbergen amphibole-bearing subcontinental lithospheric mantle (SCLM) source. In contrast, in the central part of the Lena Trough (CLT), the basalts are alkalic with relatively high Mg-number (60–65), high SiO 2 (51·0–51·6 wt %), Al 2 O 3 (18·1–18·4 wt %), Na 2 O (4·0–4·2 wt %), K 2 O (1·0–1·6 wt %), K 2 O/TiO 2 (0·6–0·9) and (La/Sm) PM (1·4–1·8), and low FeO (6·5–6·8 wt %) contents. These basalts display isotope variations with 87 Sr/ 86 Sr ranging from 0·70361 to 0·70390, 143 Nd/ 144 Nd from 0·51283 to 0·51290 ( Nd + 3·7 to +5·2), 176 Hf/ 177 Hf from 0·28313 to 0·28322 ( Hf + 11·6 to +14·9) and 206 Pb/ 204 Pb from 17·752 to 17·884, 207 Pb/ 204 Pb from 15·410 to 15·423 and 208 Pb/ 204 Pb from 37·544 to 37·670. These isotope compositions clearly distinguish the CLT lavas from those generated along the Gakkel Ridge. For the CLT lavas, involvement of a phlogopite- or amphibole- and (possibly garnet)-bearing SCLM source component is proposed. Owing to SCLM contamination along the entire length of the Lena Trough, we classify the Lena Trough as an ocean–continent transition boundary. Magmatism similar to that observed in the Lena Trough would be expected to occur wherever ocean spreading initiates.

Description: The Western Alpine Sesia–Lanzo Zone (SLZ) is a sliver of eclogite-facies continental crust exhumed from mantle depths in the hanging wall of a subducted oceanic slab. Eclogite-facies felsic and basic rocks sampled across the internal SLZ show different degrees of retrograde metamorphic overprint associated with fluid influx. The weakly deformed samples preserve relict eclogite-facies mineral assemblages that show partial fluid-induced compositional re-equilibration along grain boundaries, brittle fractures and other fluid pathways. Multiple fluid influx stages are indicated by replacement of primary omphacite by phengite, albitic plagioclase and epidote as well as partial re-equilibration and/or overgrowths in phengite and sodic amphibole, producing characteristic step-like compositional zoning patterns. The observed textures, together with the map-scale distribution of the samples, suggest open-system, pervasive and reactive fluid flux across large rock volumes above the subducted slab. Thermodynamic modelling indicates a minimum amount of fluid of 0·1–0·5 wt % interacting with the wall-rocks. Phase relations and reaction textures indicate mobility of K, Ca, Fe and Mg, whereas Al is relatively immobile in these medium-temperature–high-pressure fluids. Furthermore, the thermodynamic models show that recycling of previously fractionated material, such as in the cores of garnet porphyroblasts, largely controls the compositional re-equilibration of the exhumed rock body.

Description: Entrainment of xenoliths and their consequent assimilation are key processes in modifying the crystallization kinetics and magma dynamics of conduit systems. Here, an integrated textural and mineral chemical study of the evolution of a suite of gabbroic inclusions within a set of sheet intrusions from the Ardnamurchan Central Complex, NW Scotland, is presented. The key findings are as follows: (1) the host magma sheets and inclusions are not cognate; (2) there are microstructural and mineral chemical similarities between the gabbroic inclusions and the textures and mineralogy of the major Hypersthene Gabbro on Ardnamurchan; (3) orientations of magnetic fabrics within the host sheet groundmass and within the gabbroic inclusions are virtually identical. Field observations suggest that the inclusions were derived from the Hypersthene Gabbro and were entrained in a few laterally restricted magma segments that subsequently coalesced with inclusion-free segments into continuous sheets. Using Stokes’ Law and adaptations thereof, we calculate that the magma within the inclusion-free segments behaved as a Newtonian fluid, with a potential settling velocity of 〈0·028 m s –1 . In contrast, the presence of gabbro inclusions probably modified the magma dynamics to Bingham-like behaviour. We infer that this variation in the magma rheology of separate segments continued after coalescence and internally partitioned the magma sheet, preventing lateral mixing and inclusion transport.

Description: The Varberg–Torpa charnockite–granite association (Varberg, SW Sweden) consists of the magmatic Varberg charnockite (1399 ± 6 Ma) and the Torpa granite (1380 ± 12 Ma). The Torpa granite is both continuous and, based on its whole-rock geochemistry, synmagmatic with the Varberg charnockite. The granite body also contains a number of charnockite inliers. P – T estimation using garnet–clinopyroxene and orthopyroxene–clinopyroxene Fe–Mg exchange thermometry and garnet–orthopyroxene–plagioclase–quartz barometry gives temperatures and pressures (750–850°C; 800–850 MPa) that most probably approximate the P – T conditions during emplacement of the charnockite compared with a lower crystallization temperature (650–700°C) for the granite. The earliest recognized fluid inclusions in both the granite and charnockite consist of H 2 O–CO 2 mixtures (H 2 O volume fraction 0·2–0·7). Fluid inclusions in the charnockite are characterized by high CO 2 densities (up to 1·0 g cm – 3 ; 40–90% bulk CO 2 ), of probable magmatic origin, and are best preserved in garnet, plagioclase, and fluorapatite (in order of decreasing CO 2 densities), and sometimes also in clinopyroxene. Fluid inclusions with the highest CO 2 densities (1·08–1·10 g cm – 3 ) are found in quartz ( T h –31 to –36°C) and may have originated under high P – T conditions during emplacement and cooling of the charnockite. Magmatic fluids in the granite correspond to aqueous–carbonic inclusions with an estimated bulk composition (mol %) of H 2 O 73%, CO 2 25%, NaCl 2%. The salinity of the solutes in the granite (typically 14–20 wt % NaCl-eq.) is generally higher than for the charnockite (0–8 wt % NaCl-eq.). Field, petrographic, mineralogical, geochemical, and fluid inclusion evidence indicates that, compared with the H 2 O-rich granite, the magma responsible for the charnockite had a preponderance of CO 2 over H 2 O, which lowered the H 2 O activity in the melt, stabilizing ortho- and clinopyroxene. This evidence also supports the idea that the granite and charnockite were derived from a common source magma (most probably a fluid-rich basalt at the base of the crust) as a result of fractional crystallization.

Description: We performed crystallization experiments at 2–3 GPa at 700–950°C on basaltic and pelitic lithologies with added water and sulphur to constrain the factors controlling sulphur behaviour in subduction zones and how it may have varied through geological time. The resulting hydrous silicic melts have up to 20 times more dissolved sulphur (up to 1 wt %) than at 0·2–0·4 GPa, when moderately oxidized conditions prevail. Such high solubilities appear to result from the combined effects of enhanced solubility of water in high-pressure silicate melts (10–20 wt % H 2 O), which acts to decrease silica activity, and oxidizing conditions. The results confirm previous findings that high sulphur contents in silicate melts do not necessarily require iron-rich compositions, suggesting instead that sulphur–water complexes play a fundamental role in sulphur dissolution mechanisms in iron-poor silicic melts, in agreement with recent spectroscopic data. The experimental melts reproduce Phanerozoic slab-derived magmas, in particular their distinct Ca- and Mg-rich composition. The results also show that sulphur increases the degree of melting of basalt lithologies. Hence, we suggest that subducted slabs will preferentially melt where sulphur is present in abundance and that the variability in arc magma sulphur output reflects, in part, the vagaries of sulphur distribution in the slab source. In contrast, comparison with the composition of Archean felsic rocks suggests that, in the early Earth, much less sulphur was present in subducted slabs, in agreement with a number of independent lines of evidence showing that the Archean ocean, hence the hydrothermally altered subducted Archean oceanic crust, was considerably poorer in sulphur than at present. Volcanic degassing of sulphur was thus probably much weaker during the Archean than in Proterozoic–Phanerozoic times.

Description: Anak Krakatau is a basaltic andesite cone that has grown following the famous caldera-forming 1883 eruption of Krakatau. It breached sea level in 1927 and since the 1950s has been growing at an average rate of ~8 cm a week. We present new major and trace element data combined with whole-rock 18 O, Sr and Nd isotope data for 1883, 1993 and 2002 Krakatau eruptive products and the surrounding crust. Bombs erupted from Anak Krakatau during 2002 contain frothy metasedimentary and plutonic xenoliths that show variable degrees of thermal metamorphism, plastic deformation and partial melting. Contact-metamorphic minerals such as cordierite and tridymite in metasedimentary xenoliths are consistent with high-temperature metamorphism and incorporation at mid- to upper-crustal depth. Energy-constrained assimilation and fractional crystallization modelling of whole-rock data suggests that the Anak Krakatau magmas have a genetic relationship with the 1883 eruption products. The geochemical impact of crustal contaminants on whole-rock compositions is apparently small, and we conclude that low levels of assimilation of a quartzo-feldspathic sediment are recorded in Anak Krakatau magmas. Plagioclase phenocrysts from the 2002 eruption exhibit disequilibrium textures and complex compositional zoning, however, and are also isotopically variable with a total range in 87 Sr/ 86 Sr of 0·7043–0·7048 as determined by in situ laser ablation inductively coupled plasma mass spectrometry. This suggests that although shallow crustal assimilation appears to have had a limited effect on whole-rock chemistry, a complex late-stage differentiation history is recorded within the magma’s cargo of crystals and xenoliths.

Description: The shallow crustal Torres del Paine Intrusive Complex in southern Patagonia offers an opportunity to understand the chemical evolution and timing of crystallization processes in shallow plutonic rocks. It is characterized by hornblende-gabbros, gabbronorites, monzodiorites and granitic plutonic rocks. The exceptional exposure of the intrusion permits the identification of two structurally and petrographically different zones. Layered gabbronorite, olivine-bearing pyroxene–hornblende gabbronorite and monzodiorite forming vertical sheets and stocks in the west are referred to here as the feeder zone. These mafic rocks are in vertical contact with younger granitic rocks on their eastern border. The eastern part is a laccolith complex. It is characterized by three major units (I, II, III) of granitic rocks of over 1000 m vertical thickness; these are underlain in places by a sequence of hornblende-gabbro sills intermingled with evolved monzodiorite granite. Chilled, crenulated margins as well as flame structures between gabbroic rocks and monzodiorites suggest that the mafic sill complex remained partially molten during most of its construction. Bulk-rock major and trace element data indicate that the Paine mafic rocks follow a high-K calc-alkaline to shoshonitic differentiation trend. The parental magmas were basaltic trachyandesite liquids, with variable H 2 O and alkali contents. The majority of the feeder zone gabbronorites have high Al 2 O 3 contents and positive Eu and Sr anomalies, consistent with accumulation of plagioclase and efficient extraction of intercumulus melt. The mafic sill complex largely lacks these cumulate signatures. Comparisons of the intercumulus groundmass in the hornblende-gabbros with intra-sill dioritic stocks and pods reveal similar rare earth element patterns and trace element ratios indicating incomplete extraction of evolved interstitial liquids. The Sr, Nd and Pb isotopic compositions of the mafic and granitic rocks exhibit ranges of 87 Sr/ 86 Sr of 0·704–0·708, Nd +3·8 to –1·2, 206 Pb/ 204 Pb 18·61–18·77, 207 Pb/ 204 Pb 15·67–15·67 and 208 Pb/ 204 Pb 38·56–38·77. Crystal fractionation and assimilation–fractional crystallization modelling, combined with high-precision U–Pb dating of zircons, indicates that the western feeder zone gabbronorites are linked to the uppermost Paine granite (granite I), whereas the mafic sill complex is younger and not directly related to the voluminous granite units II and III. These results are interpreted to indicate that crystal–liquid separation is facilitated in subvertical, dynamic feeder systems whereas subhorizontal sill complexes are inefficient in separating large volumes of mafic cumulates and complementary felsic rocks.

Description: MgAl-rich sapphirine granulites (bulk X Mg 0·71–0·75) occur as boudinaged layers in migmatitic garnet–orthopyroxene–cordierite–spinel gneisses and migmatitic garnet–sillimanite metapelites in the vicinity of the c . 930–920 Ma Rogaland anorthosite–mangerite–charnockite complex, SW Norway. Investigation of the mineral reaction history of the sapphirine granulites and the surrounding paragneisses, combined with geothermobarometric calculations and constraints from pseudosections calculated in the Na 2 O–CaO–K 2 O–FeO–MgO–Al 2 O 3 –SiO 2 –H 2 O–TiO 2 (NCKFMASHT) system, indicates a clockwise P – T path that reached peak-metamorphic ultrahigh-temperature (UHT) conditions of c . 1000°C at c . 7·5 kbar by prograde heating. UHT peak metamorphism is followed by near-isothermal (ultra)high-temperature decompression to P 〈 5·5 kbar at 900–1000°C and subsequent near-isobaric cooling to 〈750–800°C at c . 5 kbar. In situ U–Pb laser ablation inductively coupled plasma mass spectrometry dating of metamorphic zircon within the sapphirine granulites yields concordant ages of 1010 ± 7 Ma and 1006 ± 4 Ma for zircon presumably formed during prograde breakdown of garnet at T 〉 850–940°C as estimated from Ti-in-zircon thermometry, suggesting that UHT metamorphism and the deduced clockwise P – T evolution is linked to regional Sveconorwegian metamorphism at c . 1010 Ma. Most of the metamorphic zircon surrounds largely resorbed inherited oscillatory zoned zircon cores ( 207 Pb/ 206 Pb apparent ages 1220–1841 Ma), testifying to the sedimentary origin of the sapphirine granulites. Epitactic growth of xenotime on metamorphic zircon at 933 ± 5 Ma is suggested to be related to crystallization of anatectic melt during post-decompressional cooling. The clockwise P – T path culminating at mid-crustal UHT conditions at c . 1010 Ma followed by (U)HT decompression is interpreted to result from collisional tectonics during the early stages of the Sveconorwegian Orogeny, followed by gravitational collapse of the mountain plateau.

Description: We investigate the time scales of magma genesis, melt evolution, crystal growth rates and magma degassing in the Erebus volcano magmatic system using measurements of 238 U– 230 Th– 226 Ra– 210 Pb– 210 Po, 232 Th– 228 Ra– 228 Th and 235 U– 231 Pa– 227 Ac. These are the first measurements of 231 Pa– 227 Ac in volcanic samples and represent the first set of data in a volcanic system to examine the entire suite of relevant 238 U, 235 U and 232 Th decay series nuclides. Our sample suite consists of 22 phonolite volcanic bombs, erupted between 1972 and 2005, and five anorthoclase megacrysts separated from bombs erupted in 1984, 1989, 1993, 2004 and 2005. The 238 U– 230 Th, 230 Th– 226 Ra and 235 U– 231 Pa systems are uniform over the 34 years examined. The anorthoclase megacrysts and phonolite glasses show complementary 226 Ra/ 230 Th disequilibria with ( 226 Ra/ 230 Th) ~40 in the anorthoclase and ~0·75 in the phonolite glass. In all samples, ( 210 Pb/ 226 Ra) is in radioactive equilibrium for both phases. In two phonolite glass samples ( 227 Ac/ 231 Pa) is unity. For the phonolite glasses ( 228 Ra/ 232 Th) is in equilibrium, whereas in the anorthoclase megacrysts it is significantly greater than unity. Instantaneous crystal fractionation, with magma residence times greater than 100 years and less than 10 kyr, can account for the measured 238 U– 230 Th– 226 Ra– 210 Pb and 235 U– 231 Pa– 227 Ac. However, the significant 228 Ra/ 232 Th disequilibria in the anorthoclase megacrysts preclude this simple interpretation. To account for this apparent discrepancy we therefore developed an open-system, continuous crystallization model that incorporates both nuclide ingrowth and decay during crystallization. This open-system model successfully reproduces all of the measured 238 U and 232 Th disequilibria and suggests that the shallow magma reservoir at Erebus is growing. The implication of this modeling is that when the time scale of crystallization is comparable with the half-life of the daughter nuclide of interest (e.g. 226 Ra) the simple isochron techniques typically used in most U-series studies can provide erroneous ages. The observation that ( 210 Pb/ 226 Ra) and ( 227 Ac/ 231 Pa) are in radioactive equilibrium suggests that the residence time of the magmas is 〉100 years. When considering the effect of 222 Rn degassing on 210 Pb/ 226 Ra, the data indicate that the majority of magma degassing is deep and long before eruption, consistent with melt inclusion data. Additionally, for the 2005 lava bomb, whose eruption date (16 December 2005) is known explicitly, 210 Po was not completely degassed from the magma at the time of eruption. Incomplete degassing of 210 Po is atypical for subaerially erupted lavas and suggests that the Erebus shallow magma degasses about 1% of its Po per day. The combined 238 U and 232 Th data further indicate that the pyroclasts ejected by Strombolian eruptions at Erebus have compositions that are close to what would be expected for a near-steady-state system, reflecting inmixing of degassed magmas, crystal fractionation, and aging.

Description: The geochemistry of pyroclasts sampled from four volcanoes along the Kermadec arc in the SW Pacific is used to investigate the genesis of silicic magmas in a young (〈2 Myr), archetypical intra-oceanic arc setting. Raoul, Macauley and Raoul SW volcanoes in the northern Kermadec arc, and Healy volcano in the southern Kermadec arc have all recently erupted dacitic to rhyolitic crystal-poor pumice. In addition to whole-rock analyses, we present a detailed study of mineral and glass chemistries to highlight the complex structure of the Kermadec magmatic systems. Major and trace element bulk-rock compositions mostly fall into relatively narrow compositional ranges, forming discrete groups by eruption for Raoul, and varying with relative crystal contents for Healy. In contrast, pumices from Macauley cover a wide range of compositions, between 66 and 72·5 wt % SiO 2 . At all four volcanoes the trace element patterns of pumice are subparallel to both those of previously erupted basalts and/or whole mafic blebs found both as discrete pyroclasts and as inclusions within pumices. Pb and Sr isotopic compositions have limited ranges within single volcanoes, but vary considerably along the arc, being more radiogenic in the southern volcanoes. Distinctive crystal populations and zonation patterns in pumices, mafic blebs and plutonic xenoliths indicate that many crystals did not grow in the evolved magmas, but are instead mixed from other sources including gabbros and hydrothermally altered tonalites. Such open-system mixing is ubiquitous at the four volcanoes. Oxygen isotope compositions of both phenocrysts (silicic origin) and xenocrysts or antecrysts (mafic origin) are typical for mantle-derived melts. Whole-rock, glass and mineral chemistries are consistent with evolved magmas being generated at each volcano through ~70–80% crystal fractionation of a basaltic parent. Our results are not consistent with silicic magma generation via crustal anatexis, as previously suggested for these Kermadec arc volcanoes. Although crystallization is the dominant process driving melt evolution in the Kermadec volcanoes, we show that the magmatic systems are open to contributions from both newly arriving melts and wholly crystalline plutonic bodies. Such processes occur in variable proportions between magma batches, and are largely reflected in small-scale chemical variations between eruption units.

Description: Here we address the question of the origin of rare nepheline-normative arc magmas through a systematic study of major, minor and trace elements in primitive olivine-hosted melt inclusions, together with literature data. The host-rocks of the inclusions are Mg-rich basalts to ankaramite lavas and scoria from several intra-oceanic volcanic arcs (Vanuatu, Lesser Antilles, Indonesian, Luzon and Aeolian arcs). The studied melt inclusions display trace element patterns typical of subduction-related calc-alkaline basalts, with variable enrichments in large ion lithophile elements and Sc (20–91 ppm), and La/Yb and Nb/Y ratios ranging from 1 to 18 and from 0·1 to 0·3, respectively. In CMAS projections, the melt inclusions delineate a trend linking two well-defined end-members, which are strongly and weakly enriched in the diopside component, respectively. The melt inclusions provide snapshots of the compositions of instantaneous melts, recording compositional diversity in the primitive magma batches, which requires mixing between melts generated by partial melting of peridotite and amphibole-bearing clinopyroxene-rich lithologies, possibly at a pressure of ~1 GPa and between 1200 and 1300°C. This hypothesis is supported by trace element modeling, and particularly by the correlation of Sc with incompatible element ratios. We propose that amphibole-bearing clinopyroxenites, occurring as cumulates at the base of the crust and/or as metasomatic veins in the upper mantle of island arcs, represent a suitable source for the Ne-normative melt inclusions variably enriched in CaO, observed in arc environments.

Description: Tephra and lava pairs from two summit eruptions ( ad 2008 and 1957) and a flank fissure eruption (~ ad 1850) are compared in terms of textures, phenocryst contents, and mineral zoning patterns to shed light on processes responsible for the shifts in eruption style during typical eruptive episodes at Volcán Llaima (Andean Southern Volcanic Zone, Chile). The mineralogy and whole-rock compositions of tephra and lavas are similar within eruptive episodes, suggesting a common magma reservoir for Strombolian paroxysms and lava effusion. The zoning profiles and textures of plagioclase record successive and discrete intrusions of volatile-rich mafic magma accompanied by mixing of these recharge magmas with the resident basaltic-andesitic crystal mushes that are commonly present at shallow levels in the Llaima system. Each recharge event destabilizes the plagioclase in equilibrium with the resident crystal mush melt and stabilizes relatively An-rich plagioclase, as is recorded by the numerous resorption zones. Lavas typically have ~15–20 vol. % more phenocrysts than the tephra. Differences in plagioclase and olivine textures and zoning, combined with different phenocryst contents, indicate that a greater volume fraction of recharge magma is present in the explosively erupted magma than in subsequent effusively erupted magma. We propose that Strombolian paroxysms at Volcán Llaima are triggered by interactions with large volume fractions of recharge magma, which decrease the bulk viscosity and increase the volatile contents of the erupted magmas, leading to the conditions required for the fragmentation of basaltic-andesite. Lava effusion ensues from reduced interactions with the recharge magma, after it has partially degassed and crystallized, thereby impeding rapid ascent. This process could be operating at other steady-state basaltic volcanoes, wherein shallow reservoirs are periodically refilled by fresh, volatile-rich magmas.

Description: The celebrated truncated T-matrix method for wave propagation models belongs to a class of the reduced basis methods (RBMs), with the parameters being incident waves and incident directions. The T-matrix characterizes the scattering properties of the obstacles independent of the incident and receiver directions. In the T-matrix method the reduced set of basis functions for representation of the scattered field is constructed analytically and hence, unlike other classes of the RBM, the T-matrix RBM avoids computationally intensive empirical construction of a reduced set of parameters and the associated basis set. However, establishing a convergence analysis and providing practical a priori estimates for reducing the number of basis functions in the T-matrix method has remained an open problem for several decades. In this work we solve this open problem for time-harmonic acoustic scattering in two and three dimensions. We numerically demonstrate the convergence analysis and the a priori parameter estimates for both point-source and plane-wave incident waves. Our approach can be used in conjunction with any numerical method for solving the forward wave propagation problem.

Description: We derive an algorithm for the adaptive approximation of solutions to parabolic equations. It is based on adaptive finite elements in space and the implicit Euler discretization in time with adaptive time-step sizes. We prove that, given a positive tolerance for the error, the adaptive algorithm reaches the final time with a space–time error between continuous and discrete solution that is below the given tolerance. Numerical experiments reveal a more than competitive performance of our algorithm ASTFEM (adaptive space–time finite element method).

Description: We propose and analyse a new family of nonconforming elements for the Brinkman problem of porous media flow. The corresponding finite element methods are robust with respect to the limiting case of Darcy flow, and the discretely divergence-free functions are in fact divergence-free. Therefore, in the absence of sources and sinks, the method is strongly mass-conservative. We also show how the proposed elements are part of a discrete de Rham complex.

Description: In this paper we investigate the superconvergence of local discontinuous Galerkin (LDG) methods for solving one-dimensional linear time-dependent fourth-order problems. We prove that the error between the LDG solution and a particular projection of the exact solution, e u , achieves th-order superconvergence when polynomials of degree k ( k ≥ 1) are used. Numerical experiments with P k polynomials, with 1 ≤ k ≤ 3, are displayed to demonstrate the theoretical results, which show that the error e u actually achieves ( k +2)th-order superconvergence, indicating that the error bound for e u obtained in this paper is suboptimal. Initial boundary value problems, nonlinear equations and solutions having singularities, are numerically investigated to verify that the conclusions hold true for very general cases.

Description: The discrete mollification method, a convolution-based filtering procedure for the regularization of illposed problems, is applied here to stabilize explicit schemes, which were first analysed by Karlsen & Risebro (2001, An operator splitting method for nonlinear convection–diffusion equations. M2AN Math. Model. Numer. Anal. 35 , 239–269) for the solution of initial value problems of strongly degenerate parabolic partial differential equations in two space dimensions. Two new schemes are proposed, which are based on directionwise and two-dimensional discrete mollification of the second partial derivatives forming the Laplacian of the diffusion function. The mollified schemes permit substantially larger time steps than the original (basic) scheme. It is proven that both schemes converge to the unique entropy solution of the initial value problem. Numerical examples demonstrate that the mollified schemes are competitive in efficiency, and in many cases significantly more efficient, than the basic scheme.

Description: A family of explicit adaptive algorithms is designed to solve nonlinear scalar one-dimensional conservation laws. Based on the Godunov scheme on a uniform grid, a first strategy uses the multiresolution analysis of the solution to design an adaptive grid that evolves in time according to the time-dependent local smoothness. The method is furthermore enhanced by a local time-stepping strategy. Both numerical schemes are shown to converge towards the unique entropy solution.

Description: In this paper the first error analyses of hybridizable discontinuous Galerkin (HDG) methods for convection–diffusion equations for variable-degree approximations and nonconforming meshes are presented. The analysis technique is an extension of the projection-based approach recently used to analyse the HDG method for the purely diffusive case. In particular, for approximations of degree k on all elements and conforming meshes, we show that the order of convergence of the error in the diffusive flux is k + 1 and that of a projection of the error in the scalar unknown is 1 for k = 0 and k + 2 for k 〉 0. When nonconforming meshes are used our estimates do not rule out a degradation of 1/2 in the order of convergence in the diffusive flux and a loss of 1 in the order of convergence of the projection of the error in the scalar variable. However, they do guarantee the optimal convergence of order k + 1 of the scalar variable.

Description: Fast multigrid solvers are considered for the linear systems arising from the bilinear finite element discretizations of second-order elliptic equations with anisotropic diffusion. Optimal convergence of Vcycle multigrid methods in the semicoarsening case and nearly optimal convergence of V-cycle multigrid method with line smoothing in the uniformly-coarsening case are established using the Xu-Zikatanov identity. Since the ‘regularity assumption’ is not used in the analysis, the results can be extended to general domains consisting of rectangles.

Description: In this paper we consider a class of incompressible viscous fluids whose viscosity depends on the shear rate and pressure. We deal with isothermal steady flow and analyse the Galerkin discretization of the corresponding equations. We discuss the existence and uniqueness of discrete solutions and their convergence to the solution of the original problem. In particular, we derive a priori error estimates, which provide optimal rates of convergence with respect to the expected regularity of the solution. Finally, we demonstrate the achieved results by numerical experiments. The fluid models under consideration appear in many practical problems, for instance, in elastohydrodynamic lubrication where very high pressures occur. Here we consider shear-thinning fluid models similar to the power-law/Carreau model. A restricted sublinear dependence of the viscosity on the pressure is allowed. The mathematical theory concerned with the self-consistency of the governing equations has emerged only recently. We adopt the established theory in the context of discrete approximations. To our knowledge, this is the first analysis of the finite element method for fluids with pressure-dependent viscosity. The derived estimates coincide with the optimal error estimates established recently for Carreau-type models, which are covered as a special case.

Description: Discrete maximum principles (DMPs) are established for finite element approximations of systems of nonlinear parabolic partial differential equations with mixed boundary and interface conditions. The results are based on an algebraic DMP for suitable systems of ordinary differential equations.

Description: The adaptive cubic regularization algorithm described in Cartis et al. (2009, Adaptive cubic regularisation methods for unconstrained optimization. Part I: motivation, convergence and numerical results. Math. Program. , 127 , 245–295; 2010, Adaptive cubic regularisation methods for unconstrained optimization. Part II: worst-case function- and derivative-evaluation complexity [online]. Math. Program. , DOI: 10.1007/s10107-009-0337-y) is adapted to the problem of minimizing a nonlinear, possibly nonconvex, smooth objective function over a convex domain. Convergence to first-order critical points is shown under standard assumptions, without any Lipschitz continuity requirement on the objective's Hessian. A worst-case complexity analysis in terms of evaluations of the problem's function and derivatives is also presented for the Lipschitz continuous case and for a variant of the resulting algorithm. This analysis extends the best-known bound for general unconstrained problems to nonlinear problems with convex constraints.

Description: This work is intended to provide a convenient tool for the mathematical analysis of a particular kind of finite volume approximation which can be used, for instance, in the context of nonlinear and/or anisotropic diffusion operators in three dimensions (3D). Following the so-called DDFV (discrete duality finite volume) approach developed by Hermeline (1998, Une méthode de volumes finis pour les équations elliptiques du second ordre. C. R. Math. Acad. Sci. Paris , 326 , 1433–1436 (in French); 2000, A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys. , 160 , 481–499) and by Domelevo & Omnès (2005, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. M2AN Math. Model. Numer. Anal. , 39 , 1203–1249) in 3D, we consider a ‘double’ covering T of a 3D domain by a rather general primal mesh and by a well-chosen ‘dual’ mesh. The associated discrete divergence operator div  T is obtained by the standard finite volume approach. A simple and consistent discrete gradient operator T is defined by a local affine interpolation that takes into account the geometry of the double mesh. Under mild geometrical constraints on the choice of the dual volumes, we show that –div  T and T are linked by the ‘discrete duality property’, which is an analogue of the integration-by-parts formula. The primal mesh need not be conformal, and its interfaces can be general polygons. We give several numerical examples for anisotropic linear diffusion problems; good convergence properties are observed. The sequel, Andreianov et al. (2011a, On 3D DDFV discretization of gradient and divergence operators. II. Discrete functional analysis tools and applications to degenerate parabolic problems. HAL preprint available at http://hal.archives-ouvertes.fr/hal-00567342 ) to this paper will summarize some key discrete functional analysis tools for DDFV schemes and give applications to proving convergence of DDFV schemes for several nonlinear degenerate parabolic partial differential equations.

Description: This work is concerned with the numerical implementation of the discrepancy principle for nonsmooth Tikhonov regularization for linear inverse problems. First, some theoretical properties of the solutions to the discrepancy equation, i.e., uniqueness and upper bounds, are discussed. Then, the idea of Padé approximation is exploited for designing model functions with model parameters iteratively updated. Two algorithms are proposed for its efficient numerical realization, i.e., a two-parameter algorithm based on model functions and a quasi-Newton method, and their convergence properties are briefly discussed. Numerical results for four nonsmooth models are presented to demonstrate the accuracy of the principle and to illustrate the efficiency and robustness of the proposed algorithms.

Description: This paper is devoted to the convergence analysis of the upwind finite volume scheme for the initialand boundary-value problems associated with the linear transport equation in any dimension, on general unstructured meshes. We are particularly interested in the case where the initial and boundary data are in L and the advection vector field has low regularity properties, namely L 1 (]0, T [, ( W 1,1 ()) d ), with suitable assumptions on its divergence. In this general framework, we prove uniform in time strong convergence in L p (), with p 〈 +, of the approximate solution towards the unique weak solution of the problem as well as the strong convergence of its trace. The proof relies, in particular, on the Friedrichs' commutator argument, which is classical in the renormalized solutions theory. Note that this result remains valid if the data are suitably approximated in L 1 . This is nothing but the discrete counterpart of the nice compactness properties deduced from the renormalized solution theory. We conclude with some numerical experiments showing that the convergence rate seems to be 1/2, as in the case of smoother advection fields, but this is still an open question up to now.

Description: In this paper, we study the existence, uniqueness and regularity properties of solutions for the nonstandard Volterra integral equation . We then present a collocation method to solve this equation, and analyse the convergence and superconvergence of piecewise polynomial collocation approximations. We also illustrate the theoretical results by extensive numerical experiments.

Description: In this paper we consider the discretization error in space and time of an H 1 gradient flow for an energy integral where the energy density is given by the sum of a double-well potential term and a bending energy term. We show that the problem is equivalent to a nonlinear heat equation with nonlocal nonlinearity and adapt the standard error analysis theory developed for the nonlinear heat equation to our case. In doing so we bound the discretization error in terms of the mesh size and time step as well as energy parameters. In particular, we carefully track how the size of the bending energy affects the error bounds.

Description: A local convergence analysis of Newton's method for finding a singularity of a differentiable vector field defined on a complete Riemannian manifold, based on the majorant principle, is presented in this paper. This analysis provides a clear relationship between the majorant function, which relaxes the Lipschitz continuity of the derivative, and the vector field under consideration. It also allows us to obtain the optimal convergence radius and the biggest range for the uniqueness of the solution and to unify some previously unrelated results.

Description: For analytic functions we study the kernel of the remainder terms of Gaussian quadrature rules with respect to Bernstein–Szego weight functions where 0〈 α 〈 β , β != 2 α , | |〈 β – α , and whose denominator is an arbitrary polynomial of exact degree 2 that remains positive on [–1, 1]. The subcase α =1, β =2/(1+ ), –1〈 〈0 and =0 has been considered recently by Spalevíc, M. M. & Praníc, M. S. ((2010) Error bounds of certain Gaussian quadrature formulae. J. Comput. Appl. Math., 234 , 1049–1057). The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective error bounds for the corresponding Gaussian quadratures. The approach we use in this paper, which is different from the one adopted in Spalevíc, M. M. & Praníc, M. S. ((2010) Error bounds of certain Gaussian quadrature formulae. J. Comput. Appl. Math., 234 , 1049–1057), ensures that the actual conditions for determining the locations on the elliptic contours where the modulus of the kernel attains its maximum value are approximated very precisely.