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  • CR: 5.17  (43)
  • Springer  (43)
  • American Geophysical Union
  • Blackwell Publishing Ltd
  • Nature Publishing Group
  • PANGAEA
  • 1980-1984  (43)
  • 1925-1929
  • 1983  (19)
  • 1980  (24)
  • 1925
Collection
Keywords
Publisher
  • Springer  (43)
  • American Geophysical Union
  • Blackwell Publishing Ltd
  • Nature Publishing Group
  • PANGAEA
Years
  • 1980-1984  (43)
  • 1925-1929
Year
  • 1
    Electronic Resource
    Electronic Resource
    Numerische Lösung gewöhnlicher Differentialgleichungen mit Splinefunktionen (1980)
    Springer
    Computing 25 (1980), S. 317-335 
    Details
    ISSN: 1436-5057
    Keywords: AMS (MOS): 65L65 ; CR: 5.17 ; AMS (MOS): 65L65 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Abstract In this paper a general procedure to obtain spline approximations for the solutions of initial value problems for ordinary differential equations is presented. Several well-known spline approximation methods are included as special cases. It is common practice to partition the interval for which the initial value problem is defined into equidistant subintervals and to construct successively the spline approximation; thereby the spline function has to satisfy certain conditions at the knots. In the general procedure presented here additional knots are admitted in every subinterval. At these points which need not be equally spaced the spline approximation has to fulfill analogous conditions as at the original knots. Convergence and divergence theorems are proved; especially the influence of the additional knots on convergence and divergence of the method is investigated.
    Notes: Zusammenfassung In dieser Arbeit wird ein allgemeines Verfahren zur Erzeugung von Splineapproximationen für die Lösungen von Anfangswertproblemen bei gewöhnlichen Differentialgleichungen vorgestellt. Einige der bekannten Spline-approximationsmethoden sind als Spezialfälle enthalten. Eine gängige Vorgehensweise besteht darin, das Intervall, über dem das Anfangswertproblem gegeben ist, in äquidistante Teilintervalle zu zerlegen und dann sukzessive die Splineapproximation zu definieren. Hierbei wird gefordert, daß die Spline-approximation in den Knoten gewisse Bedingungen erfüllt. Bei dem hier betrachteten allgemeinen Verfahren werden in den einzelnen Teilintervallen noch zusätzliche Zwischenknoten eingeführt. In diesen Punkten, die nicht äquidistant sein müssen, werden für die Splineapproximation analoge Bedingungen wie in den Hauptknoten vorgeschrieben. Konvergenz- und Divergenzsätze werden bewiesen, insbesondere wird der Einfluß der Zwischenknoten auf Konvergenz und Divergenz des Verfahrens untersucht.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02285227
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  • 2
    Electronic Resource
    Electronic Resource
    A semi-implicit mid-point rule for stiff systems of ordinary differential equations (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 373-398 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The paper introduces a new semi-implicit extrapolation method especially designed for the numerical solution of stiff systems of ordinary differential equations. The existence of a quadratic asymptotic expansion in terms of the stepsize is shown. Moreover, the new discretization is analyzed in the light of well-known stability models. The efficiency of the new integrator is clearly demonstrated by solving a series of challenging test problems including real life examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01418331
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  • 3
    Electronic Resource
    Electronic Resource
    Maximum-norm stability and error estimates in Galerkin methods for parabolic equations in one space variable (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 345-371 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Maximum-norm stability and error estimates of best approximation and nonsmooth data types are derived for the approximate solution of a parabolic equation in one space variable, using the continuous in time Galerkin method based on piecewise polynomial approximating functions on a quasi-uniform mesh.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01418330
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  • 4
    Electronic Resource
    Electronic Resource
    On the uniform power-boundedness of a family of matrices and the applications to one-leg and linear multistep methods (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 1-13 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We study the difference equations obtained when a linear multistep method is applied to the scalar test equationdy/dt=λy and constant stepsizeh. LetS be the region of the absolute stability of the method, and letD be a closed subset ofS (on the Riemann sphere $$\mathbb{C}$$ ). It is shown that the solutions of these difference equations are bounded forn≧0, uniformly for λh∈D.S is itself closed in $$\mathbb{C}$$ iff ∂S is free of cusps. The question is studed by means of contractivity analysis and a matrix theorem, derived from the matrix theorem of Kreiss.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400914
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  • 5
    Electronic Resource
    Electronic Resource
    Order and stepsize control in extrapolation methods (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 399-422 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The paper presents a new theory for joint order and stepsize control in extrapolation methods. This theory defines a locally optimal order that can be determined along any trajectory to be computed. In addition, Shannon's information theory is applied to derive some ideal convergence model that is expected to describe the behavior of an extrapolation method over a large set of test problems. Extensive numerical comparisons document a drastic acceleration in stiff integration and a mild acceleration in non-stiff integration by the new device. Moreover, a significant increase in reliability, robustness, and portability of the extrapolation codes is achieved.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01418332
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  • 6
    Electronic Resource
    Electronic Resource
    Split linear multistep methods for the numerical integration of stiff differential systems (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 299-310 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A new approach to the problem of numerically integrating stiff differential systems is described. In this approach a linear multistep method (the basic method) is split into a kind of predictor-corrector scheme, where the predictor is also implicit. If this splitting is done in an appropriate manner, the modified method has considerably better stability properties than the basic method. As a result, splitting methods are particularly useful for problems where conventional integration methods experience stability difficulties. In particular some highly stable split linear multistep methods based on backward differentiation formulae are derived and a highly stable variable step implementation is proposed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389575
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  • 7
    Electronic Resource
    Electronic Resource
    A boundary element method for a two-dimensional interface problem in electromagnetics (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 311-322 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65 N 30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper presents Galerkin approximations for solutions of two dimensional interface problems by solving corresponding boundary integral equations. These are obtained by simple layer potential operators only. Due to the strong ellipticity of the integral equations the Galerkin procedure converges with optimal order. Smoothness of the given data implies high convergence rates for the layers.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389576
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  • 8
    Electronic Resource
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    OnD-stability andB-stability (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 349-357 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper theD-stability properties of some classes of perturbed collocation methods are investigated. Among others we show the implicit Runge-Kutta methods described by Ehle-Chipman to beD-stable. We also give implicit Runge-Kutta methods that are bothD-stable andA-stable but notB-stable. Finally we indicate in what senseB-stability is stronger thanA-stability together withD-stability. The results of this paper are based on recent results of Nørsett and Wanner on perturbed collocation methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389579
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  • 9
    Electronic Resource
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    On the integration of stiff systems of O.D.E.s using extended backward differentiation formulae (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 235-246 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A class of extended backward differentiation formulae suitable for the approximate numerical integration of stiff systems of first order ordinary differential equations is derived. An algorithm is described whereby the required solution is predicted using a conventional backward differentiation scheme and then corrected using an extended backward differentiation scheme of higher order. This approach allows us to developL-stable schemes of order up to 4 andL(α)-stable schemes of order up to 9. An algorithm based on the integration formulae derived in this paper is illustrated by some numerical examples and it is shown that it is often superior to certain existing algorithms.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396701
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  • 10
    Electronic Resource
    Electronic Resource
    An approximation theorem for second-order evolution equations (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 127-142 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 34G05, 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A class of approximation schemes of arbitrary accuracy, generated by a two-step recurrence relation, is devised for evolution equations of the second order. The schemes are effected via a specially constructed family of rational approximations to cos τ for τ≧0 and yield computationally efficient methods for systems of second-order ordinary differential equations and semidiscrete approximations for initial-boundary value problems for second-order hyperbolic equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396311
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  • 11
    Electronic Resource
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    Méthodes multipas pour des équations paraboliques non linéaires (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 143-162 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65M15, 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We study the error due to the discretization in time of a nonlinear parabolic problem by a multistep method. Error estimates are obtained if the method is of the orderp (p〉1) and stronglyA(Θ)-stable $$\left( {0〈 \Theta〈 \frac{\pi }{2}} \right)$$ . The method is also applied to the Navier-Stokes equations in two dimensions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396312
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  • 12
    Electronic Resource
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    Enlarging the domain of convergence for multiple shooting by the homotopy method (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 231-240 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L10 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The homotopy method is a frequently used technique in overcoming the local convergence nature of multiple shooting. In this paper sufficient conditions are given that guarantee the homotopy process to be feasible. The results are applicable to a class of two-point boundary value problems. Finally, the numerical solution of two practical problems arising in physiology is described.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396319
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  • 13
    Electronic Resource
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    Une méthode multipas implicite-explicite pour l'approximation des équations d'évolution paraboliques (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 257-276 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) ; 65M10 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Résumé Considérons une équation d'évolution parabolique linéaire associée à un opérateur linéaireA(t) dépendant du tempst. Nous développons dans cet article une méthode de discrétisation, basée sur les méthodes linéaires à pas multiples, traitant de manière implicite une partie de l'opérateurA(t) indépendante du temps, l'autre partie est traitée de manière explicite. Nous étudions la stabilité et la convergence de cette méthode.
    Notes: Summary Let us consider a linear parabolic equation which is associated with a time dependent operatorA(t). In this paper, we present a method, which is founded on linear multistep methods, which discretize a time-independent part of the operatorA(t) in an implicit way, and the other part in an explicit way. We study stability and convergence for this method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396412
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  • 14
    Electronic Resource
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    Mixed finite elements in ℝ3 (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 315-341 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) ; 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We present here some new families of non conforming finite elements in ℝ3. These two families of finite elements, built on tetrahedrons or on cubes are respectively conforming in the spacesH(curl) andH(div). We give some applications of these elements for the approximation of Maxwell's equations and equations of elasticity.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396415
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  • 15
    Electronic Resource
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    A mixed finite element approximation of the Navier-Stokes equations (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 381-404 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We study in this paper the convergence of a new mixed finite element approximation of the Navier-Stokes equations. This approximation uses low order Lagrange elements, leads to optimal order of convergence for the velocity and the pressure, and induces an efficient numerical algorithm for the solution of this problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01399007
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  • 16
    Electronic Resource
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    Error analysis for Laplace transform —Finite element solution of hyperbolic equations (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 55-62 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The finite element method with Laplace transform of time variable is proposed for the solution of hyperbolic equations. Error estimates in Hardy spaces of functions with values in Sobolev spaces are derived. Due to the isometric isomorphism of Hardy spaces with weighted Hilbert spaces these estimates are valid also for original formulations of hyperbolic equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396305
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  • 17
    Electronic Resource
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    Smoothing the extrapolated midpoint rule (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 165-175 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65LO5 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The extrapolated midpoint rule is a popular way to solve the initial value problem for a system of ordinary differential equations. As originally formulated by Gragg, the results are smoothed to remove the weak instability of the midpoint rule. It is shown that this smoothing is not necessary. A cheaper smoothing scheme is proposed. A way to exploit smoothing to increase the robustness of extrapolation codes is formulated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01390211
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  • 18
    Electronic Resource
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    On calculating the eigenvalues of the finite Hill's differential equation (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 255-279 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L15 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Some methods for evaluating the characteristic exponents in connection with Newton's iteration are applied for solving the eigenvalue problem related to the finite Hill's differential equation or, in particular, Mathieu's equation. By using these methods a high accuracy is achieved, furthermore a complete error analysis, which yields rather realistic error bounds, is possible.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01390215
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  • 19
    Electronic Resource
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    Monotone methods for periodic solutions of second order scalar functional differential equations (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 15-30 
    Details
    ISSN: 0945-3245
    Keywords: AMS: 65L10 ; 34C25 ; 34K10 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary An iterative method is presented which starting from a lower or from an upper periodic solution, provides a monotone sequence converging to a periodic solution of (1). With some restrictions on the growth off, the method extends to functional differential equations of type (1′). Two numerical examples with an “a posteriori” error analysis are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400915
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  • 20
    Electronic Resource
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    Approximation of pseudodifferential operators in absorbing boundary conditions for hyperbolic equations (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 51-64 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 65N99 ; 35L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Engquist and Majda [3] proposed a pseudodifferential operator as asymptotically valid absorbing boundary condition for hyperbolic equations. (In the case of the wave equation this boundary condition is valid at all frequencies.) Here, least-squares approximation of the symbol of the pseudodifferential operator is proposed to obtain differential operators as boundary conditions. It is shown that for the wave equation this approach leads to Kreiss well-posed initial boundary value problems and that the expectation of the reflected energy is lower than in the case of Taylor- and Padé-approximations [3, 4]. Numerical examples indicate that this method works even more effectively for hyperbolic systems. The least-squares approach may be used to generate the boundary conditions automatically.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400917
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  • 21
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    On the condition number of boundary integral operators for the exterior Dirichlet problem for the Helmholtz equation (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 77-95 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N99 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Brakhage and Werner, Leis and Panich suggested to reduce the exterior Dirichlet boundary value problem for the Helmholtz equation to an integral equation of the second kind which is uniquely solvable for all frequencies by seeking the solution in the form of a combined double- and single-layer potential. We present an analysis of the appropriate choice of the parameter coupling the double- and single-layer potential in order to minimize the condition number of the integral operator.
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    URL: http://dx.doi.org/10.1007/BF01400919
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  • 22
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    On multi-grid methods for variational inequalities (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 65-76 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65 N 20 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider here a general class of algorithms for the numerical solution of variational inequalities. A convergence proof is given and in particular a multi-grid method is described. Numerical results are presented for the finite-difference discretization of an obstacle problem for minimal surfaces
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    URL: http://dx.doi.org/10.1007/BF01400918
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  • 23
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    Die maximale Konsistenzordnung von Differenzenapproximationen nichtnegativer Art (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 119-123 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N10 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Many difference methods for the numerical solution of elliptic boundary value problems lead to systems of linear equations whose matrices areM-matrices and which therefore have nonnegative inverses. In this paper it is shown, that these difference methods are at most consistent of second order.
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    URL: http://dx.doi.org/10.1007/BF01400922
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  • 24
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    On conforming finite element methods for the inhomogeneous stationary Navier-Stokes equations (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 173-194 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider the stationary Navier-Stokes equations, written in terms of the primitive variables, in the case where both the partial differential equations and boundary conditions are inhomogeneous. Under certain conditions on the data, the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed. A conforming finite element method is presented and optimal estimates for the error of the approximate solution are proved. In addition, the convergence properties of iterative methods for the solution of the discrete nonlinear algebraic systems resulting from the finite element algorithm are given. Numerical examples, using an efficient choice of finite element spaces, are also provided.
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    URL: http://dx.doi.org/10.1007/BF01395310
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  • 25
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    Contractivity in the numerical solution of initial value problems (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 271-290 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65J10, 65L20, 65M10 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Consider a linear autonomous system of ordinary differential equations with the property that the norm |U(t)| of each solutionU(t) satisfies |U(t)|≦|U(0)| (t≧0). We call a numerical process for solving such a system contractive if a discrete version of this property holds for the numerical approximations. A givenk-step method is said to be unconditionally contractive if for each stepsizeh〉0 the numerical process is contractive. In this paper a general theory is given which yields necessary and sufficient conditions for unconditional contractivity. It turns out that unconditionally contractive methods are subject to an order barrierp≦1. Further the concept of a contractivity threshold is studied, which makes it possible to compare the contractivity behaviour of methods with an orderp〉1 as well. Most theoretical results in this paper are formulated for differential equations in arbitrary Banach spaces. Applications are given to numerical methods for solving ordinary as well as partial differential equations.
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    URL: http://dx.doi.org/10.1007/BF01389573
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  • 26
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    Stability of multistep-methods on variable grids (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 359-377 
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    ISSN: 0945-3245
    Keywords: AMS 65L05 ; 65L07 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper is concerned with the stability of multistep methods for ordinary initial-value problems on grids with variable mesh-sizes. A necessary and sufficient condition for stability is given from which generalizations of recent results by Gear et al. and by Zlatev can be obtained as special cases. As an application the stability of the variable BDF-formulas is treated.
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    URL: http://dx.doi.org/10.1007/BF01389580
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  • 27
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    Norm bounds for rational matrix functions (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 379-389 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L20 ; 15A60 ; 65M20 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary If the field of values of a matrixA is contained in the left complex halfplaneH and a functionf mapsH into the unit disc then ∥f(A)∥2≦1 by a theorem of J.v. Neumann. We prove a theorem of this type, only the field of values ofA is used for functions which are absolutely bounded by one in only part ofH. An extension can be used to show norm-stability of single step methods for stiff differential equations. The results are applicable among others to several subdiagonal Padé approximations which are notA-stable.
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    URL: http://dx.doi.org/10.1007/BF01389581
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    Die mehrstellenformeln für den Laplaceoperator (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 171-187 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Difference methods for the numerical solution of linear partial differential equations may often be improved by using a weighted right hand side instead of the original right hand side of the differential equation. Difference formulas, for which that is possible, are called “Mehrstellenformeln’ or Hermitian formulas. In this paper the Hermitian formulas for the approximation of Laplace's operator are characterized by a very simple condition. We prove, that in two-dimensional case for a Hermitian formula of ordern at leastn+3 discretization points are necessary. We give examples of such optimal formulas of arbitrary high-order.
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    URL: http://dx.doi.org/10.1007/BF01396058
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    Effective computation of periodic orbits and bifurcation diagrams in delay equations (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 457-467 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05, 65Q05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary By employing a numerical method which uses only rather classical tools of Numerical Analysis such as Newton's method and routines for ordinary differential equations, unstable periodic solutions of differential-difference equations can be computed. The method is applied to determine bifurcation diagrams with backward bifurcation.
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    URL: http://dx.doi.org/10.1007/BF01403681
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    A finite element method for the Tricomi problem (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 13-20 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): primary 65M05 ; 65M10 ; 65M15 ; secondary: 35M05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider the numerical solution of the Tricomi problem. Using a weak formulation based on different spaces of test and trial functions, we construct a new Galerkin procedure for the Tricomi problem. Existence, uniqueness, and uniform stability of the approximate solution is proven, and a priori error bounds are given.
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    URL: http://dx.doi.org/10.1007/BF01396366
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    Deferred corrections using uncentered end formulas (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 21-33 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65B05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The trapezoidal rule with deferred corrections using uncentered end formulas is shown to converge. While the proof technique is more specialized than the standard asymptotic expansion approach, it has some advantages. In addition to providing a more complete theoretical justification for current implementations of deferred corrections with the trapezoidal rule, the approach given here will hopefully apply for several other discretization methods.
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    Unconditionally stable explicit methods for parabolic equations (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 57-68 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; 65M20 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper discussesrational Runge-Kutta methods for stiff differential equations of high dimensions. These methods are explicit and in addition do not require the computation or storage of the Jacobian. A stability analysis (based onn-dimensional linear equations) is given. A second orderA 0-stable method with embedded error control is constructed and numerical results of stiff problems originating from linear and nonlinear parabolic equations are presented.
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    URL: http://dx.doi.org/10.1007/BF01396370
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    On the design of high order exponentially fitted formulae for the numerical integration of stiff systems (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 253-266 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65LO5 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Single step exponentially fitted integration formulae of orders 4 and 6 are derived. The final approximation to the required solution is obtained via a linear combination of asymptotically less accurate solutions obtained using conventional implicit integration formulae. This linear combination is performed in such a way that the final integration formula, as well as having an increased order of accuracy, integrates a certain pre-determined ordinary differential equation exactly. The algorithms developed are illustrated by means of some numerical examples.
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    URL: http://dx.doi.org/10.1007/BF01396654
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    Exact bounds for the solution branches of nonlinear eigenvalue problems (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 29-40 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N25 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A method is constructed which yields a strip containing the full solution sets of nonlinear eigenvalue problems of the formu=λTu. The strip can be narrowed iteratively, and the method applies for both stable and unstable branches. Its high degree of accuracy is demonstrated by numerical examples. In particular, a lower bound is given for the critical value at which criticality is lost in the thermal ignition problem for the unit ball.
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    URL: http://dx.doi.org/10.1007/BF01463996
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    Analysis of finite element methods for second order boundary value problems using mesh dependent norms (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 41-62 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper presents a new approach to the analysis of finite element methods based onC 0-finite elements for the approximate solution of 2nd order boundary value problems in which error estimates are derived directly in terms of two mesh dependent norms that are closely ralated to theL 2 norm and to the 2nd order Sobolev norm, respectively, and in which there is no assumption of quasi-uniformity on the mesh family. This is in contrast to the usual analysis in which error estimates are first derived in the 1st order Sobolev norm and subsequently are derived in theL 2 norm and in the 2nd order Sobolev norm — the 2nd order Sobolev norm estimates being obtained under the assumption that the functions in the underlying approximating subspaces lie in the 2nd order Sobolev space and that the mesh family is quasi-uniform.
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    Splineapproximationen von beliebigem Defekt zur numerischen Lösung gewöhnlicher Differentialgleichungen. Teil III (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 143-154 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L65 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In the first part [1] a general procedure is presented to obtain polynomial spline approximations for the solutions of initial value problems for ordinary differential equations; furthermore a divergence theorem is proved there. Sufficient conditions for convergence of the method are given in the second part [2]. The remaining case which has not been considered in [1] and [2] is treated in the present paper. In this special case the procedure is equivalent to an unstable two-step method with special initial values; nevertheless, convergence can be proved. Finally,A 0-stability of the method as well as the influence of rounding errors are investigated.
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    URL: http://dx.doi.org/10.1007/BF01396056
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    Iterative solutions of nonlinear initial value differential equations in Chebyshev series using Lie series (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 1-13 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 34A45, 34A50, 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper Lie series are presented in Chebyshev form and applied to the iterative solution of initial value problems in differential equations. The resulting method, though algebraically complicated, is of theoretical interest as a generalisation of Taylor series methods and iterative Chebyshev methods. The theory of the method is discussed and the solutions of some simple scalar equations are analysed to illustrate the behaviour of the process.
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    URL: http://dx.doi.org/10.1007/BF01463994
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    A method of characteristics solving the initial-boundary value problem of a hyperbolic differential equation of second order (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 217-234 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65M25, 35L20 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The ALGOL-procedure1 char2 presented in this paper can be applied to the initial or initial-boundary value problem of a quasilinear hyperbolic differential equation of second order. A method of characteristics is combined with extrapolation to the limit. Thus, the results are very accurate. The same accuracy can also be obtained if the initial values are only piecewise smooth.
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    URL: http://dx.doi.org/10.1007/BF01396700
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    Finite dimensional approximation of nonlinear problems (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 1-25 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65 N 30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We begin in this paper the study of a general method of approximation of solutions of nonlinear equations in a Banach space. We prove here an abstract result concerning the approximation of branches of nonsingular solutions. The general theory is then applied to the study of the convergence of two mixed finite element methods for the Navier-Stokes and the von Kármán equations.
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    URL: http://dx.doi.org/10.1007/BF01395985
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    Convergence of the nonconforming Wilson element for arbitrary quadrilateral meshes (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 33-52 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A modified variational formulation, recently introduced by Taylor, Beresford and Wilson for solving second order problems, using the nonconforming Wilson element is here analysed. It is shown that the Patch Test is satisfied and that stresses and displacements are respectively first and second order accurate for arbitrary quadrilateral meshes.
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    URL: http://dx.doi.org/10.1007/BF01395987
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    Numerical solution of steady-state porous flow free boundary problems (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 73-98 
    Details
    ISSN: 0945-3245
    Keywords: AMS: 35R35, 65P05, 76S05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A new numerical method is used to solve stationary free boundary problems for fluid flow through porous media. The method also applies to inhomogeneous media, and to cases with a partial unsaturated flow.
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    URL: http://dx.doi.org/10.1007/BF01395990
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    A finite element lumped mass scheme for solving eigenvalue problems of circular arches (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 267-290 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper, we present a finite element lumped mass scheme for eigenvalue problems of circular arch structures, and give error estimates for the approximation. They assert that approximate eigenvalues and eigenfuctions converge to the exact ones. Some numerical examples are also given to illustrate our results.
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    A note onB-stability of Runge-Kutta methods (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 319-331 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS) ; 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Burrage and Butcher [1, 2] and Crouzeix [4] introduced for Runge-Kutta methods the concepts ofB-stability,BN-stability and algebraic stability. In this paper we prove that for any irreducible Runge-Kutta method these three stability concepts are equivalent.
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    URL: http://dx.doi.org/10.1007/BF01396658
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