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  • Articles  (45)
  • CR: 5.17  (45)
  • 1975-1979  (45)
  • 1970-1974
  • Mathematics  (45)
  • Architecture, Civil Engineering, Surveying
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  • Articles  (45)
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  • Mathematics  (45)
  • Architecture, Civil Engineering, Surveying
  • 1
    Electronic Resource
    Electronic Resource
    Stability properties of variable stepsize variable formula methods (1978)
    Springer
    Numerische Mathematik 31 (1978), S. 175-182 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65 L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary When variable stepsize variable formula methods (VSVFM's) are used in the solution of systems of first order differential equations instability arises sometimes. Therefore it is important to find VSVFM's whose zerostability properties are not affected by the choice of both the stepsize and the formula. The Adams VSVFM's are such methods. In this work a more general class of methods which contains the Adams VSVFM's is discussed and it is proved that the zero-stability of the class is not affected by the choice of the stepsize and of the formula.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01397474
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  • 2
    Electronic Resource
    Electronic Resource
    A-priori error estimates of Galerkin backward differentiation methods in time-inhomogeneous parabolic problems (1978)
    Springer
    Numerische Mathematik 30 (1978), S. 369-383 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65L05, 65M10, 65M15, 65M20, 65N15, 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Backward differentiation methods up to orderk=5 are applied to solve linear ordinary and partial (parabolic) differential equations where in the second case the space variables are discretized by Galerkin procedures. Using a mean square norm over all considered time levels a-priori error estimates are derived. The emphasis of the results lies on the fact that the obtained error bounds do not depend on a Lipschitz constant and the dimension of the basic system of ordinary differential equations even though this system is allowed to have time-varying coefficients. It is therefore possible to use the bounds to estimate the error of systems with arbitrary varying dimension as they arise in the finite element regression of parabolic problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01398506
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  • 3
    Electronic Resource
    Electronic Resource
    Error estimates for the finite element solution of variational inequalities (1978)
    Springer
    Numerische Mathematik 31 (1978), S. 1-16 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We study the mixed finite element approximation of variational inequalities, taking as model problems the so called “obstacle problem” and “unilateral problem”. Optimal error bounds are obtained in both cases.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396010
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  • 4
    Electronic Resource
    Electronic Resource
    One-step methods of any order for ordinary differential equations with discontinuous right-hand sides (1978)
    Springer
    Numerische Mathematik 31 (1978), S. 131-152 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The right-hand sides of a system of ordinary differential equations may be discontinuous on a certain surface. If a trajectory crossing this surface shall be computed by a one-step method, then a particular numerical analysis is necessary in a neighbourhood of the point of intersection. Such an analysis is presented in this paper. It shows that one can obtain any desired order of convergence if the method has an adequate order of consistency. Moreover, an asymptotic error theory is developed to justify Richardson extrapolation. A general one-step method is constructed satisfying the conditions of the preceding theory. Finally, a simplified Newton iteration scheme is used to implement this method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01397472
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  • 5
    Electronic Resource
    Electronic Resource
    Splineapproximationen von beliebigem Defekt zur numerischen Lösung gewöhnlicher Differentialgleichungen. Teil I (1979)
    Springer
    Numerische Mathematik 32 (1979), S. 147-157 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS) ; 65L65 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In 1967, F.R. Loscalzo und T.D. Talbot presented a procedure for obtaining polynomial spline approximations of defect one for solutions of the initial value problem for first order ordinary differential equations. This method is generalized and investigated for spline approximations of arbitrary defect. The results are analogous to those of the defect one. In the first part of this paper the divergence problem is treated. The convergent procedures are investigated in a following second part of this paper.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01404871
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  • 6
    Electronic Resource
    Electronic Resource
    Sur les elements finis rationnels de Wachspress (1979)
    Springer
    Numerische Mathematik 32 (1979), S. 247-270 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D05, 65N30, 41A20 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Résumé Ce travail est consacré à l'étude d'éléments finis rationnels de Wachspress (cf. [15]) sur un quadrilatère. Après avoir étudié la construction et les propriétés de ces éléments finis, on obtient une évaluation de l'erreur d'interpolation correspondante.
    Notes: Summary This paper is devoted to the study of Wachspress's rational finite elements (cf. [15]) over a quadrilateral. After studying construction and properties of these finite elements, we get an estimate of the corresponding interpolation error.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01397000
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  • 7
    Electronic Resource
    Electronic Resource
    Splineapproximationen von beliebigem Defekt zur numerischen Lösung gewöhnlicher Differentialgleichungen. Teil II (1979)
    Springer
    Numerische Mathematik 32 (1979), S. 343-358 
    Details
    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 [5] a general procedure is presented to obtain polynomial spline approximations of arbitrary defect for the solution of the initial value problem of ordinary differential equations. The essential result is a divergence theorem in dependence of the polynomial degree and the defect of the spline functions. In this second part the convergent procedures are investigated and two convergence theorems are proved. Furthermore the question is treated, whether the convergent procedures are appropriate for the numerical solution of stiff equations. The paper is finished by a convergence theorem for a procedure producing spline approximations in a natural way by the discrete approximations of an arbitrary difference method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01397006
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  • 8
    Electronic Resource
    Electronic Resource
    Efficient algorithms for solving tensor product finite element equations (1978)
    Springer
    Numerische Mathematik 31 (1978), S. 49-61 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65N20 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary There are currently several highly efficient methods for solving linear systems associated with finite difference approximations of Poisson's equation in rectangular regions. These techniques are employed to develop both direct and iterative methods for solving the linear systems arising from the use ofC 0 quadratic orC 1 cubic tensor product finite elements.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396013
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  • 9
    Electronic Resource
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    Acceleration of convergence for finite element solutions of the Poisson equation (1979)
    Springer
    Numerische Mathematik 33 (1979), S. 43-53 
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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 Letu h be the finite element solution to−Δu=f with zero boundary conditions in a convex polyhedral domain Ω. Fromu h we calculate for eachz∈Ω and |α|≦1 an approximationu h −α (z) toD α u(z) with |D α u(z)−u h −α (z)|=O(h 2k−2) wherek is the order of the finite elements. The same superconvergence order estimates are obtained also for the boundary flux. We need not work on a regular mesh but we have to compute averages ofu h where the diameter of the domain of integration must not depend onh.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396494
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  • 10
    Electronic Resource
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    Extrapolation to the limit for numerical solutions of hyperbolic equations (1977)
    Springer
    Numerische Mathematik 28 (1977), S. 455-474 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65B05, 65M25, 65M99 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The application of extrapolation to the limit requires the existence of an asymptotic expansion in powers of the step size. In this paper one-and multi-step methods for the solution of hyperbolic systems of first order are considered. Conditions are formulated that ensure the asymptotic expansion. Methods of characteristics for quasilinear systems with two independent variables are included in this presentation. If a rectangular grid is used, also non-quasilinear systems are admissible. The main part of this paper deals with initial value problems. But it is shown that in some exceptional cases asymptotic expansions hold for initial-boundary problems, too.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01404347
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  • 11
    Electronic Resource
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    Stabile Mehrschichtverfahren für parabolische Evolutionsgleichungen (1977)
    Springer
    Numerische Mathematik 29 (1977), S. 65-82 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65M20 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The line method discussed in [19] is improved by considering several time slices. It is applied to parabolic initial-value problems, and with techniques similar to [16] it is proved that the root conditions of Dahlquist [5] and Widlund [17] are sufficient for stability. Stable methods up to order 6 are given and illustrated by numerical calculations.
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    URL: http://dx.doi.org/10.1007/BF01389314
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  • 12
    Electronic Resource
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    On the order of composite multistep methods for ordinary differential equations (1978)
    Springer
    Numerische Mathematik 29 (1978), S. 381-396 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper, a general class ofk-step methods for the numerical solution of ordinary differential equations is discussed. It is shown that methods with order of consistencyq have order of convergence (q+1) if a very simple condition is satisfied. This result gives a new aspect to previous results of Spijker; it also serves as a starting point for a new theory of cyclick-step methods, completing an approach of Donelson and Hansen. It facilitates the practical determination of high-order cyclick-step methods, especially of stiffly stable,k-step methods.
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    URL: http://dx.doi.org/10.1007/BF01432876
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  • 13
    Electronic Resource
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    On the order of iterated defect correction (1978)
    Springer
    Numerische Mathematik 29 (1978), S. 409-424 
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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 In a recent article [2] Frank and Überhuber define and motivate the method of iterated defect correction for Runge-Kutta methods. They prove a theorem on the order of that method using the theory of asymptotic expansions. In this paper we give similar results using the theory of Butcher series (see [4]). Our proofs are purely algebraic. We don't restrict our considerations to Runge-Kutta methods, but we admit arbitrary linear one-step methods. At the same time we consider more general defect functions as in [2].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01432878
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  • 14
    Electronic Resource
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    Zur Stabilität von Differenzenverfahren für Systeme linearer gewöhnlicher Randwertaufgaben (1978)
    Springer
    Numerische Mathematik 29 (1978), S. 209-226 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L 10 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper we give a simple stability theory for finite difference approximations to linear ordinary boundary value problems. In particular we consider stability with respect to a maximum norm including all difference quotients up to the order of the differential equation. It is shown that stability in this sense holds if and only if the principal part of the differential equation is discretized in a “stable way”. This last property is characterized by root conditions which we prove to be satisfied for some classes of finite difference schemes. Our approach simplifies and generalizes some known results of the literature where Sobolev norms or merely the maximum norm are used.
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    URL: http://dx.doi.org/10.1007/BF01390339
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  • 15
    Electronic Resource
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    Transformation of three-dimensional regions onto rectangular regions by elliptic systems (1978)
    Springer
    Numerische Mathematik 29 (1978), S. 397-407 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 31-04, 65N05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A transformation method is developed which may be used to solve various types of boundary value problems on three-dimensional regions with an arbitrary boundary. The implementation of the method is illustrated in the solution of a potential flow problem. All computations are performed on a cubic mesh in a rectangular region.
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    URL: http://dx.doi.org/10.1007/BF01432877
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  • 16
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    The defect correction principle and discretization methods (1978)
    Springer
    Numerische Mathematik 29 (1978), S. 425-443 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65J05, 65L99 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Recently, a number of closely related techniques for error estimation and iterative improvement in discretization algorithms have been proposed. In this article, we expose the common structural principle of all these techniques and exhibit the principal modes of its implementation in a discretization context.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01432879
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  • 17
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    Einschließungsaussagen bei Differentialoperatoren zweiter Ordnung durch punktweise Ungleichungen (1978)
    Springer
    Numerische Mathematik 30 (1978), S. 93-101 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L10 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider non-inverse positive differential operators of the second order and show that error estimates can be obtained through methods of inverse positivity alone. This means a significant simplification if the operator has only one negative eigenvalue. In addition we show how to obtain Range-Domain implications for operators with mixed boundary conditions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01403909
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  • 18
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    A Taylor series method for the numerical solution of two-point boundary value problems (1978)
    Springer
    Numerische Mathematik 31 (1978), S. 359-375 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 65L10 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary For the numerical solution of two-point boundary value problems a shooting algorithm based on a Taylor series method is developed. Series coefficients are generated automatically by recurrence formulas. The performance of the algorithm is demonstrated by solving six problems arising in nonlinear shell theory, chemistry and superconductivity.
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    URL: http://dx.doi.org/10.1007/BF01404566
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  • 19
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    A computational solution for a Matrix Riccati differential equation (1979)
    Springer
    Numerische Mathematik 32 (1979), S. 271-279 
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    ISSN: 0945-3245
    Keywords: AMS(MOS) 49.04 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper is concerned with the solution of the finite time Riccati equation. The solution to the Riccati equation is given in terms of the partition of the transition matrix. Matrix differential equations for the partition of the transition matrix are derived and are solved using computational methods. Examples illustrating the method are presented and the computational algorithms are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01397001
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  • 20
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    Convergence of multistep methods for Volterra functional differential equations (1979)
    Springer
    Numerische Mathematik 32 (1979), S. 307-332 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65Q05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper deals with the convergence of nonstationary quasilinear multistep methods with varying step, used for the numerical integration of Volterra functional differential equations. A Perron type condition (appearing in the differential equations theory) is imposed on the increment function. This gives a generalization of some results of Tavernini ([19–21]).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01397004
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  • 21
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    On the mixed finite element approximation for the buckling of plates (1979)
    Springer
    Numerische Mathematik 33 (1979), S. 195-210 
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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 We consider the mixed finite element method for the buckling problem of the thin plate by using piecewise linear polynomials. We give error estimates for the approximate eigenvalues and the eigenfunctions.
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    URL: http://dx.doi.org/10.1007/BF01399554
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  • 22
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    A numerical method to boundary value problems for second order delay differential equations (1979)
    Springer
    Numerische Mathematik 33 (1979), S. 69-75 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 34B15 ; 65L10 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In the first part of this paper we are dealing with theoretical statements and conditions which lead to existence and uniqueness of the solution of a nonlinear boundary value problem with delay. Next we apply this method successfully to a numerical example. The computations have been carried out at the computer Siemens 4004. The data obtained are presented in two tables.
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    URL: http://dx.doi.org/10.1007/BF01396496
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  • 23
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    The dependence of critical parameter bounds on the monotonicity of a Newton sequence (1979)
    Springer
    Numerische Mathematik 33 (1979), S. 291-301 
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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 new method is proposed for the inclusion of the critical parameter λ* of some convex operator equationu=λTu (appearing e.g. in thermal explosion theory). It is based on the fact that for a fixed λ Newton's method starting with a suitable subsolution is not monotonically if and only if λ〉λ*. Several numerical examples arising from nonlinear boundary value problems illustrate the efficiency of the method.
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    URL: http://dx.doi.org/10.1007/BF01398645
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  • 24
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    Numerische Behandlung von Verzweigungsproblemen bei gewöhnlichen Differentialgleichungen (1979)
    Springer
    Numerische Mathematik 32 (1979), S. 17-29 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L10 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We present a new method for the numerical solution of bifurcation problems for ordinary differential equations. It is based on a modification of the classical Ljapunov-Schmidt-theory. We transform the problem of determining the nontrivial branch bifurcating from the trivial solution into the problem of solving regular nonlinear boundary value problems, which can be treated numerically by standard methods (multiple shooting, difference methods).
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    URL: http://dx.doi.org/10.1007/BF01397647
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  • 25
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    Numerical computation of branch points in ordinary differential equations (1979)
    Springer
    Numerische Mathematik 32 (1979), S. 51-68 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 65L10 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper deals with the computation of branch points in ordinary differential equations. A direct numerical method is presented which requires the solution of only one boundary value problem. The method handles the general case of branching from a nontrivial solution which is a-prioriunknown. A testfunction is proposed which may indicate branching if used in continuation methods. Several real-life problems demonstrate the procedure.
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    URL: http://dx.doi.org/10.1007/BF01397649
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  • 26
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    On the convergence of an inverse iteration method for nonlinear elliptic eigenvalue problems (1979)
    Springer
    Numerische Mathematik 32 (1979), S. 69-74 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N25, 35J60, 49D20 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In recent years, a group of inverse iteration type algorithms have been developed for solving nonlinear elliptic eigenvalue problems in plasma physics [4]. Although these algorithms have been very successful in practice, no satisfactory theoretical justification of convergence has been available. The present paper fills this gap and proves for a large class of such problems and a simple version of such algorithms that linear convergence to a local maximum of a certain potential is obtained.
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    URL: http://dx.doi.org/10.1007/BF01397650
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  • 27
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    A 0-Stability and stiff stability of Brown's multistep multiderivative methods (1979)
    Springer
    Numerische Mathematik 32 (1979), S. 167-181 
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    ISSN: 0945-3245
    Keywords: AMS(MOS) ; 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Brown [1] introducedk-step methods usingl derivatives. Necessary and sufficient conditions forA 0-stability and stiff stability of these methods are given. These conditions are used to investigate for whichk andl the methods areA 0-stable. It is seen that for allk andl withk≦1.5 (l+1) the methods areA 0-stable and stiffly stable. This result is conservative and can be improved forl sufficiently large. For smallk andl A 0-stability has been determined numerically by implementing the necessary and sufficient condition.
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    URL: http://dx.doi.org/10.1007/BF01404873
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  • 28
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    La méthode de Kikuchi appliquée aux équations de von Karman (1979)
    Springer
    Numerische Mathematik 32 (1979), S. 209-232 
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    ISSN: 0945-3245
    Keywords: AMS(MOS) ; 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Resumé Le but de cet article est l'étude de l'approximation numérique des solutions non-triviales des équations de Von Karman pour le flambage d'une plaque mince encastrée. S'inspirant de la méthode de Kikuchi pour des problèmes quasi-linéaires du second ordre, on propose une méthode itérative d'éléments finis qui donne des approximations des solutions de norme «petite» qui bifurquent de la solution triviale au voisinage d'une valeur propre simple du problème linéarisé. On démontre la convergence et on obtient des estimations de l'erreur.
    Notes: Summary The aim of this article is to study the numerical approximation of non-trivial solutions of the Von Karman equations for the buckling of a thin elastic clamped plate. Following Kikuchi's method for second order quasilinear problems, we propose an iterative finite element method which produces approximations of non-trivial solutions of “small” norm which bifurcate from the trivial solution near simple eigenvalues of the linearised problem. The convergence is proved and error estimates are obtained.
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    URL: http://dx.doi.org/10.1007/BF01404876
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  • 29
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    Equilibrium finite elements for the linear elastic problem (1979)
    Springer
    Numerische Mathematik 33 (1979), S. 367-383 
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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 a class of equilibrium finite element methods for elasticity problems. The approximate stresses satisfy the equilibrium equations but the symmetry of the stress tensor is relaxed. Optimal error bounds for the stresses and numerical examples are given.
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    URL: http://dx.doi.org/10.1007/BF01399320
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    Extrapolation methods for dynamic partial differential equations (1978)
    Springer
    Numerische Mathematik 29 (1978), S. 269-285 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 65 M 99 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Several extrapolation procedures are presented for increasing the order of accuracy in time for evolutionary partial differential equations. These formulas are based on finite difference schemes in both the spatial and temporal directions. One of these schemes reduces to a Runge-Kutta type formula when the equations are linear. On practical grounds the methods are restricted to schemes that are fourth order in time and either second, fourth or sixth order in space. For hyperbolic problems the second order in space methods are not useful while the fourth order methods offer no advantage over the Kreiss-Oliger method unless very fine meshes are used. Advantages are first achieved using sixth order methods in space coupled with fourth order accuracy in time. The averaging procedure advocated by Gragg does not increase the efficiency of the scheme. For parabolic problems severe stability restrictions are encountered that limit the applicability to problems with large cell Reynolds number. Computational results are presented confirming the analytic discussions.
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    URL: http://dx.doi.org/10.1007/BF01389212
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    The method of lines for Poisson's equation with nonlinear or free boundary conditions (1978)
    Springer
    Numerische Mathematik 29 (1978), S. 329-344 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65M20 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The method of lines is used to solve Poisson's equation on an irregular domain with nonlinear or free boundary conditions. The partial differential equation is approximated by a system of second order ordinary differential equations subject to multi-point boundary conditions. The system is solved with an SOR iteration which employs invariant imbedding for each one dimensional problem. An application of the method to a boundary control problem and to a free surface problem arising in electrochemical machining is described. Finally, some theoretical convergence results are presented for a model problem with radiative boundary conditions on fixed boundaries.
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    URL: http://dx.doi.org/10.1007/BF01389216
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    High order methods for the numerical integration of ordinary differential equations (1978)
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    Numerische Mathematik 30 (1978), S. 385-409 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65 L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary High order implicit integration formulae with a large region of absolute stability are developed for the approximate numerical integration of both stiff and non-stiff systems of ordinary differential equations. The algorithms derived behave essentially like one step methods and are demonstrated by direct application to certain particular examples.
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    URL: http://dx.doi.org/10.1007/BF01398507
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    Higher order methods for a singularly perturbed problem (1978)
    Springer
    Numerische Mathematik 30 (1978), S. 267-279 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L10 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A finite element method using piecewise polynomials of degree ≦k is used to approximate the problem εu″+u′=f, ε〉0 a small parameter. A very irregular mesh is used. On this mesh error estimates of order0(h k+1) are obtained uniformly in ε,h the maximum stepsize, fork≧2. The condition number of the system of linear equations one has to solve in order to get the approximation is estimated. Extension of the results to more complicated problems is briefly indicated. Finally, a numerical example is given.
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    URL: http://dx.doi.org/10.1007/BF01411843
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    Sur laB-stabilité des méthodes de Runge-Kutta (1979)
    Springer
    Numerische Mathematik 32 (1979), S. 75-82 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Résumé Dans cet article, nous modifions légèrement la définition de laB-stabilité donnée par J.C. Butcher [1] afin qu'elle s'applique à une plus large classe d'équations différentielles et nous donnons des caractérisations simples de cette propriété.
    Notes: Summary In this paper, we slightly modify the definition ofB-stability of Butcher [1], so as to cover a wider class of differential equations, and we give simple characterizations of this property.
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    URL: http://dx.doi.org/10.1007/BF01397651
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    On the stability properties of Brown's multistep multiderivative methods (1978)
    Springer
    Numerische Mathematik 30 (1978), S. 25-38 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Brown introducedk-step methods usingl derivatives. We investigate for whichk andl the methods are stable or unstable. It is seen that to anyl the method becomes unstable fork large enough. All methods withk≦2(l+1) are stable. Fork=1,2,..., 18 there exists aλ k such that the methods are stable for anyl ≧λ k and unstable for anyl 〈λ k . Theλ k are given.
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    URL: http://dx.doi.org/10.1007/BF01403904
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    Some equilibrium finite element methods for two-dimensional elasticity problems (1978)
    Springer
    Numerische Mathematik 30 (1978), S. 103-116 
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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 some equilibrium finite element methods for two-dimensional elasticity problems. The stresses and the displacements are approximated by using piecewise linear functions. We establishL 2-estimates of orderO(h 2) for both stresses and displacements.
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    URL: http://dx.doi.org/10.1007/BF01403910
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    An analysis of a mixed finite element method for the Navier-Stokes equations (1979)
    Springer
    Numerische Mathematik 33 (1979), S. 235-271 
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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 simple mixed finite element method is developed to solve the steady state, incompressible Navier-Stokes equations in a neighborhood of an isolated—but not necessarily unique—solution. Convergence is established under very mild restrictions on the triangulation, and, when the solution is sufficiently smooth, optimal error bounds are obtained.
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    URL: http://dx.doi.org/10.1007/BF01398643
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    Liapunov functions and error bounds for approximate solutions of ordinary differential equations (1978)
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    Numerische Mathematik 30 (1978), S. 411-414 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L99 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary It is shown that Liapunov functions may be used to obtain error bounds for approximate solutions of systems of ordinary differential equations. These error bounds may reflect the behaviour of the error more accurately than other bounds.
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    URL: http://dx.doi.org/10.1007/BF01398508
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    A note on the approximation of mildly nonlinear Dirichlet problems by finite differences (1979)
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    Numerische Mathematik 33 (1979), S. 303-313 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N20 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In [9], Simpson proved some theorems concerning the approximation of mildly nonlinear Dirichlet problems −Δu=f (x, u) inD, u=0 on ∂D by finite differences. The assumptionsf(x, 0)≧0 andf u(x, u)〉0 in [9] have turned out to be unnecessarily restrictive and are eliminated in this paper. On the other hand, we considered it necessary to make the smoothness conditions forD slightly more stringent irrespective of the conditions imposed onf.
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    On a mixed finite element approximation of the Stokes problem (I) (1979)
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    Numerische Mathematik 33 (1979), S. 397-424 
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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 study in this paper a new mixed finite element approximation of the Stokes' problem in the velocity pressure formulation. This approximation which is based on a new variational principle allows the use of low order Lagrange elements and leads to optimal order of convergence for the velocity and the pressure. Iterative and direct methods for the solution of the approximate problems will be discussed in a forthcoming paper.
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    URL: http://dx.doi.org/10.1007/BF01399323
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    Direct and inverse error estimates for finite elements with mesh refinements (1979)
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    Numerische Mathematik 33 (1979), S. 447-471 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N15 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The finite element method is used to solve a second order elliptic boundary value problem on a polygonal domain. Mesh refinements and weighted Besov spaces are used to obtain optimal error estimates and inverse theorems.
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    URL: http://dx.doi.org/10.1007/BF01399326
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    Lösung von gewöhnlichen Differentialgleichungen mit nichtlinearen Splines (1979)
    Springer
    Numerische Mathematik 33 (1979), S. 323-338 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Summary We consider the technique of using nonlinear splines to solve the initial value problem of ordinary differential equations. It is known, for example, that generalized rational splines with variable exponents yield good approximations to the exact solution in the neighborhood of a singularity. In the case of polynomial splines, convergence results may be derived by demonstrating the equivalence of the method to linear multistep methods. This sort of analysis has been done by many authors. In this paper we treat the nonlinear case and are able to prove convergence by directly estimating the local errors at interior knots. Some computational examples are given which illustrate the power of the method near a singularity.
    Notes: Zusammenfassung In dieser Arbeit werden nichtlineare Splines zur Lösung von Anfangswertaufgaben bei gewöhnlichen Differentialgleichungen herangezogen. In der Nähe von Singularitäten besitzen z.B. verallgemeinerte rationale Splines mit variablen Exponenten gute Approximationseigenschaften. Bei Polynomsplines können Konvergenzaussagen hergeleitet werden, indem Äquivalenz dieser Verfahren mit gewissen linearen Mehrschrittverfahren gezeigt wird. In dieser Arbeit behandeln wir den nichtlinearen Fall, indem wir die lokalen Fehler in den Knoten direkt verfolgen. Einige numerische Beispiele zeigen die Güte dieser Verfahren insbesondere bei solchen Lösungen, die sehr steil anwachsen oder sogar im betrachteten Intervall singulär werden.
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    On patched variational principles with application to elliptic and mixed elliptic-hyperbolic problems (1977)
    Springer
    Numerische Mathematik 28 (1977), S. 259-271 
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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 Variational principles are important tools for the approximate solution of boundary-value problems. There are many types of variational principles, and each has its advantages and disadvantages. In this paper we show how to use a combination of variational principles, each for a given subregion of the underlying region of space, so as to best utilize the chief benefits of the individual principles. Such a patched principle is particularly useful in solving transonic flow problems, where we use different principles in the elliptic and hyperbolic regions. We present the results of some numerical experiments for the Tricomi problem. These seem to indicate that our patched principle, when used in conjunction with the finite element method, leads to accuracy which is second-order in the mesh spacing, as compared to the standard numerical methods of solving this problem, which are only first-order.
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    URL: http://dx.doi.org/10.1007/BF01389967
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    Fehlerabschätzung bei nichtlinearen Differentialungleichungen mit Hilfe linearer Differentialungleichungen (1977)
    Springer
    Numerische Mathematik 28 (1977), S. 393-405 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 35-10, 35-15, 35-19, 35-42 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Ifu andv satisfy a sufficiently regular partial differential equation, it has been common knowledge for at least fifty years that the mean-value theorem gives a linear equation, with variable coefficients, foru-v. The main idea in the following note is to replace the latter by a one-parameter family of linear inequalities, each of which has constant coefficients. This procedure applies to a considerable variety of problems, but is developed here only for second-order elliptic equations in bounded or unbounded regions. A number of specific examples are included, some of which are so highly nonlinear as to seem almost intractable. Nevertheless, the method has little subtlety or depth. Such advantages as it may have lie rather in the fact that the proofs are simple, and the results easy to use.
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    URL: http://dx.doi.org/10.1007/BF01404343
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    Extrapolation applied to certain discretization methods solving the initial value problem for hyperbolic differential equations (1977)
    Springer
    Numerische Mathematik 28 (1977), S. 121-142 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): primary: 65M99 ; 65M25 ; secondary: 65B05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The procedurediwiex presented in this paper provides an approximate solution to Cauchy's initial value problem for general hyperbolic systems of first order. The procedurecharex can be applied to the initial value problem for a hyperbolic system of quasi-linear differential equations. This second method is a kind of method of characteristics. It produces a solution for the whole domain of determinancy. Both procedures use extrapolation to the limit.
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    URL: http://dx.doi.org/10.1007/BF01394448
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