Uniform expansions for a class of finite difference schemes for elliptic boundary value problems

Author:
Harry Munz

Journal:
Math. Comp. **36** (1981), 155-170

MSC:
Primary 65N05

MathSciNet review:
595048

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Abstract: For a class of finite difference schemes for the Dirichlet problem on a bounded region , the existence of uniform expansions of the approximate solution for meshlength is shown. The results also improve error bounds which Pereyra, Proskurowski, and Widlund obtained with respect to certain discrete -norms.

**[1]**J. H. Bramble and B. E. Hubbard,*A theorem on error estimation for finite difference analogues of the Dirichlet problem for elliptic equations*, Contributions to Differential Equations**2**(1963), 319–340. MR**0152134****[2]**J. H. Bramble and B. E. Hubbard,*Approximation of derivatives by finite difference methods in elliptic boundary value problems*, Contributions to Differential Equations**3**(1964), 399–410. MR**0166935****[3]**Philippe G. Ciarlet,*Discrete maximum principle for finite-difference operators*, Aequationes Math.**4**(1970), 338–352. MR**0292317****[4]**Lothar Collatz,*The numerical treatment of differential equations. 3d ed*, Translated from a supplemented version of the 2d German edition by P. G. Williams. Die Grundlehren der mathematischen Wissenschaften, Bd. 60, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR**0109436****[5]**H. Münz,*Differenzenverfahren für elliptische Randwertaufgaben mit verbesserter Randinterpolation*, Diplomarbeit, Universität Tübingen, 1978. (Unpublished.)**[6]**Victor Pereyra,*Iterated deferred corrections for nonlinear operator equations*, Numer. Math.**10**(1967), 316–323. MR**0221760****[7]**Victor Pereyra, Wlodzimierz Proskurowski, and Olof Widlund,*High order fast Laplace solvers for the Dirichlet problem on general regions*, Math. Comp.**31**(1977), no. 137, 1–16. MR**0431736**, 10.1090/S0025-5718-1977-0431736-X**[8]**Richard S. Varga,*Matrix iterative analysis*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR**0158502****[9]**David M. Young,*Iterative solution of large linear systems*, Academic Press, New York-London, 1971. MR**0305568**

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DOI:
https://doi.org/10.1090/S0025-5718-1981-0595048-7

Article copyright:
© Copyright 1981
American Mathematical Society