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  • 1
    Electronic Resource
    Electronic Resource
    A posteriori error estimation for fully discrete hierarchic models of elliptic boundary value problems on thin domains (1998)
    Springer
    Numerische Mathematik 80 (1998), S. 325-362 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65N15, 65N30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. A posteriori error estimators for fully discrete hierarchic modelling on thin domains are derived and are shown to provide computable upper bounds on the discretization error and on the total error. The estimators are shown to be robust and do not degenerate as the thickness of the domain tends to zero. If the discretization part of the error is negligible, the estimator for the modelling error reduces to the one recently obtained for semi-discrete hierarchical modelling by Babuska and Schwab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050370
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  • 2
    Electronic Resource
    Electronic Resource
    A unified approach to a posteriori error estimation using element residual methods (1993)
    Springer
    Numerische Mathematik 65 (1993), S. 23-50 
    Details
    ISSN: 0945-3245
    Keywords: 65N30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations. It is shown that, by solving appropriate local residual type problems, one can obtain upper bounds on the error in the energy norm. Moreover, in the special case of adaptiveh-p finite element analysis, the estimator will also give a realistic estimate of the error. A key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems. The work extends and combines several existing methods to the case of fullh-p finite element approximation on possibly irregular meshes with, elements of non-uniform degree. As a special case, the analysis proves a conjecture made by Bank and Weiser [Some A Posteriori Error Estimators for Elliptic Partial Differential Equations, Math. Comput.44, 283–301 (1985)].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385738
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  • 3
    Electronic Resource
    Electronic Resource
    A posteriori error estimators in the finite element method (1991)
    Springer
    Numerische Mathematik 60 (1991), S. 429-463 
    Details
    ISSN: 0945-3245
    Keywords: 65N30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In the paper we develop a structured approach to thea posteriori estimation of the error in the approximation obtained via the finite element method. This aids the classification of existing estimators as well as allowing new estimators to be proposed for new situations. A class of abstract estimators for finite elements of orderp〉1 in $$\Re ^n $$ ,n=2, 3 based on exploiting the superconvergence phenomenon are analyzed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385730
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  • 4
    Electronic Resource
    Electronic Resource
    The influence and selection of subspaces for a posteriori error estimators (1996)
    Springer
    Numerische Mathematik 73 (1996), S. 399-418 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65N30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. The element residual method for a posteriori error estimation is analyzed for degree $p$ finite element approximation on quadrilateral elements. The influence of the choice of subspace used to solve the element residual problem is studied. It is shown that the resulting estimators will be consistent (or asymptotically exact) for all $p〉1$ if and only if the mesh is parallel. Moreover, even if the mesh consists of rectangles, then the estimators can be inconsistent when $p=1$ . The results provide concrete guidelines for the selection of a posteriori error estimators and establish the limits of their performance. In particular, the use of the element residual method for high orders of approximation (such as those arising in the $h$ - $p$ version finite element method) is vindicated. The mechanism behind the rather poor performance of the estimators is traced back to the basic formulation of the residual problem. The investigations reveal a deficiency in the formulation, leading, as it does, to spurious modes in the true solution of the residual problem. The recommended choice of subspaces may be viewed as being sufficient to guarantee that the spurious modes are filtered out from the approximate solution while at the same time retaining a sufficient degree of approximation to represent the true modes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050198
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  • 5
    Electronic Resource
    Electronic Resource
    Semi-discrete Galerkin approximation of the single layer equation by general splines (1998)
    Springer
    Numerische Mathematik 79 (1998), S. 157-174 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65N30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. This paper deals with a semi-discrete version of the Galerkin method for the single-layer equation in a plane, in which the outer integral is approximated by a quadrature rule. A feature of the analysis is that it does not require high precision quadrature or exceptional smoothness of the data. Instead, the assumptions on the quadrature rule are that constant functions are integrated exactly, that the rule is based on sufficiently many quadrature points to resolve the approximation space, and that the Peano constant of the rule is sufficiently small. It is then shown that the semi-discrete Galerkin approximation is well posed. For the trial and test spaces we consider quite general piecewise polynomials on quasi-uniform meshes, ranging from discontinuous piecewise polynomials to smoothest splines. Since it is not assumed that the quadrature rule integrates products of basis functions exactly, one might expect degradation in the rate of convergence. To the contrary, it is shown that the semi-discrete Galerkin approximation will converge at the same rate as the corresponding Galerkin approximation in the $H^0$ and $H^{-1}$ norms.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050335
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  • 6
    Electronic Resource
    Electronic Resource
    An additive Schwarz preconditioner for p-version boundary element approximation of the hypersingular operator in three dimensions (2000)
    Springer
    Numerische Mathematik 85 (2000), S. 343-366 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 65N55, 65N38, 65F35
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. Additive Schwarz preconditioners are developed for the p-version of the boundary element method for the hypersingular integral equation on surfaces in three dimensions. The principal preconditioner consists of decomposing the subspace into local spaces associated with the element interiors supplemented with a wirebasket space associated with the the element interfaces. The wirebasket correction involves inverting a diagonal matrix. If exact solvers are used on the element interiors then theoretical analysis shows that growth of the condition number of the preconditioned system is bounded by $(1+\log p)^2$ for an open surface and $(1+\log p)$ for a closed surface. A modified form of the preconditioner only requires the inversion of a diagonal matrix but results in a further degradation of the condition number by a factor $p(1+\log p)$ .
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110000134
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  • 7
    Electronic Resource
    Electronic Resource
    The approximation theory for the p-version finite element method and application to non-linear elliptic PDEs (1999)
    Springer
    Numerische Mathematik 82 (1999), S. 351-388 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65N15, 65N30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. Approximation theoretic results are obtained for approximation using continuous piecewise polynomials of degree p on meshes of triangular and quadrilateral elements. Estimates for the rate of convergence in Sobolev spaces $W^{m,q}(\Omega)$ , $q\in [1,\infty]$ are given. The results are applied to estimate the rate of convergence when the p-version finite element method is used to approximate the $\alpha$ -Laplacian. It is shown that the rate of convergence of the p-version is always at least that of the h-version (measured in terms of number of degrees of freedom used). If the solution is very smooth then the p-version attains an exponential rate of convergence. If the solution has certain types of singularity, the rate of convergence of the p-version is twice that of the h-version. The analysis generalises the work of Babuska and others to the case $q\not=2$ . In addition, the approximation theoretic results find immediate application for some types of spectral and spectral element methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050423
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  • 8
    Electronic Resource
    Electronic Resource
    Local a posteriori error estimators for variational inequalities (1993)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 9 (1993), S. 23-33 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Local a posteriori error estimators for finite element approximation of variational inequalities are derived. These are shown to provide upper bounds on the discretization error. Numerical examples are given illustrating the theoretical results. © 1993 John Wiley & Sons, Inc.
    Additional Material: 6 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690090104
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  • 9
    Electronic Resource
    Electronic Resource
    The performance of Bank-Weiser's error estimator for quadrilateral finite elements (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 10 (1994), S. 609-623 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The error estimator proposed by Bank and Weiser is analyzed in the case of degree p finite element approximations on quadrilateral meshes. It is shown that the scheme is asymptotically exact in the energy norm for regular solutions provided that the degree of approximation is of odd order and the elements are rectangles. Perhaps surprisingly, the hypotheses are necessary, as counterexamples show. © 1994 John Wiley & Sons, Inc.
    Additional Material: 5 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690100508
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  • 10
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    Seasonal microclimate control of calcite fabrics, stable isotopes and trace elements in modern speleothem from St Michaels Cave, Gibraltar (2010)
    In:  Geological Society Special Publication 336: 323-344.
    Details
    Publication Date: 2010-06-21
    Description: Detailed monitoring of three drip sites in New St Michael's Cave, Gibraltar, reveals a strongly coherent seasonal pattern of dripwater chemistry despite each site having significantly different flow paths and discharge patterns. Calcite saturation is closely linked to regular seasonal variations in cave air pCO2 caused by seasonally reversing ventilation driven by temperature difference between the cave interior and the air outside. A coupled model of CO2 degassing and calcite precipitation links seasonal {delta}13C variations in coexisting dripwater, cave air CO2 and speleothem calcite to large variations in pCO2 that are driven by cave ventilation. The relationships between stable isotope ratios, Sr/Ca and speleothem fabrics across annually formed calcite laminae are consistent with a degassing-calcite precipitation process in which rapid degassing controls the {delta}13C of both drip water DIC and calcite whereas a much slower rate of calcite precipitation causes seasonal cycles of Sr in a more complex manner. By demonstrating the causes of laminated speleothem fabrics plus trace element and isotope cycles in modern speleothem from a closely monitored cave, this study provides clear links between the local microclimate and the proxy record provided by speleothem geochemistry. In Gibraltar, low cave air pCO2 in summer is unusual compared to what has been revealed by cave monitoring carried out elsewhere and shows that caution is needed when linking paired speleothem fabrics to specific seasons without knowledge of local processes operating in the cave.
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