×

Asymptotic error expansion and Richardson extrapolation for linear finite elements. (English) Zbl 0594.65082

The authors consider the approximation of the classical Dirichlet problem by finite difference schemes. More precisely, the authors improve the known error expansions for the Ritz projection method. Such asymptotic expansions justify the use of Richardson extrapolation as shown in this paper.
Reviewer: P.-L.Lions

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Böhmer, K.: Asymptotic expansions for the discretization error in linear elliptic boundary value problems on general regions. Math. Z.177, 235-255 (1981) · Zbl 0451.65071 · doi:10.1007/BF01214203
[2] Bramble, J.H., Thomée, V.: Interior maximum norm estimates for some simple finite element methods. RAIRO Anal. Numer.8, 5-18 (1974) · Zbl 0301.65065
[3] Collatz, L.: The Numerical Treatment of Differential Equations. Berlin, Heidelberg, New York: Springer 1966 · Zbl 0173.17702
[4] Frehse, J., Rannacher, R.: EineL 1-Fehlerabschätzung für diskrete Grundlösungen in der Methode der finiten Elemente. Bonn. Math. Schr.89, 92-114 (1976) · Zbl 0359.65093
[5] Levine, N.: Pointwise logarithm-free error estimates for finite elements on linear triangles. Numerical Analysis Report Nr. 6, Department of Mathematics, University of Reading 1984
[6] Lin, Q., Lu, T.: Asymptotic expansions for finite element approximation of elliptic problems on polygonal domains. Sixth Int. Conf. Comput. Math. Appl. Sci. Eng., Versailles 1983
[7] Lin, Q., Wang, J.P.: Some expansions of the finite element approximation. Research Report IMS-15, Chengdu Branch of Academia Sinica 1984 · Zbl 0553.65071
[8] Lin, Q., Zhu, Q.: Asymptotic expansion for the derivative of finite elements. J.Comput. Math.2, 361-363 (1984) · Zbl 0563.65069
[9] Rannacher, R., Scott, R.: Some optimal error estimates for piecewise linear finite element approximations. Math. Comput.38, 437-445 (1982) · Zbl 0483.65007 · doi:10.1090/S0025-5718-1982-0645661-4
[10] Schatz, A.H., Wahlbin, L.B.: Interior maximum norm estimates for finite element methods. Math. Comput.31, 414-424 (1977) · Zbl 0364.65083 · doi:10.1090/S0025-5718-1977-0431753-X
[11] Schatz, A.H., Wahlbin, L.B.: Maximum norm estimates in the finite element method on plane polygonal domains Part 1. Math. Comput.32, 73-109 (1978) · Zbl 0382.65058
[12] Volkov, E.A.: Differentiability properties of solutions of boundary value problems for the Laplace equation on a polygon. Proc. Steklov Inst. Math. 127-159 (1967); Tr. Mat. Inst. Steklova77, 113-142 (1965) · Zbl 0162.16701
[13] Wasow, W.: Discrete approximations to elliptic diffential equations. Z. Angew. Math. Phys.6, 81-97 (1955) · Zbl 0064.37802 · doi:10.1007/BF01607295
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.