Summary
The elliptic Ritz projection with linear finite elements is shown to admit asymptotic error expansions on certain uniform meshes. This justifies the application of Richardson extrapolation for increasing the accuracy.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Böhmer, K.: Asymptotic expansions for the discretization error in linear elliptic boundary value problems on general regions. Math. Z.177, 235–255 (1981)
Bramble, J.H., Thomée, V.: Interior maximum norm estimates for some simple finite element methods. RAIRO Anal. Numer.8, 5–18 (1974)
Collatz, L.: The Numerical Treatment of Differential Equations. Berlin, Heidelberg, New York: Springer 1966
Frehse, J., Rannacher, R.: EineL 1-Fehlerabschätzung für diskrete Grundlösungen in der Methode der finiten Elemente. Bonn. Math. Schr.89, 92–114 (1976)
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
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
Lin, Q., Wang, J.P.: Some expansions of the finite element approximation. Research Report IMS-15, Chengdu Branch of Academia Sinica 1984
Lin, Q., Zhu, Q.: Asymptotic expansion for the derivative of finite elements. J.Comput. Math.2, 361–363 (1984)
Rannacher, R., Scott, R.: Some optimal error estimates for piecewise linear finite element approximations. Math. Comput.38, 437–445 (1982)
Schatz, A.H., Wahlbin, L.B.: Interior maximum norm estimates for finite element methods. Math. Comput.31, 414–424 (1977)
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)
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)
Wasow, W.: Discrete approximations to elliptic diffential equations. Z. Angew. Math. Phys.6, 81–97 (1955)
Author information
Authors and Affiliations
Additional information
The work of the second author was partially supported by the Gesellschaft für Mathematik und Datenverarbeitung (GMD)
Rights and permissions
About this article
Cite this article
Blum, H., Lin, Q. & Rannacher, R. Asymptotic error expansion and Richardson extranpolation for linear finite elements. Numer. Math. 49, 11–37 (1986). https://doi.org/10.1007/BF01389427
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01389427