High order local approximations to derivatives in the finite element method

Author:
Vidar Thomée

Journal:
Math. Comp. **31** (1977), 652-660

MSC:
Primary 65D25; Secondary 65L10

DOI:
https://doi.org/10.1090/S0025-5718-1977-0438664-4

MathSciNet review:
0438664

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Abstract: Consider the approximation of the solution *u* of an elliptic boundary value problem by means of a finite element Galerkin method of order *r*, so that the approximate solution satisfies . Bramble and Schatz (*Math. Comp.*, v. 31, 1977, pp. 94-111) have constructed, for elements satisfying certain uniformity conditions, a simple function such that in the interior. Their result is generalized here to obtain similar superconvergent order interior approximations also for derivatives of *u*.

**[1]**J. H. BRAMBLE, J. A. NITSCHE & A. H. SCHATZ, "Maximum-norm interior estimates for Ritz-Galerkin methods,"*Math. Comp.*, v. 29, 1975, pp. 677-688. MR**0398120 (53:1975)****[2]**J. H. BRAMBLE & A. H. SCHATZ, "Higher order local accuracy by averaging in the finite element method,"*Math. Comp.*, v. 31, 1977, pp. 94-111. MR**0431744 (55:4739)****[3]**J. H. BRAMBLE, A. H. SCHATZ, V. THOMÉE & L. B. WAHLBIN, "Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations,"*SIAM J. Numer. Anal.*(To appear.) MR**0448926 (56:7231)****[4]**PH. BRENNER, V. THOMÉE & L. B. WAHLBIN,*Besov Spaces and Applications to Difference Methods for Initial Value Problems*, Lecture Notes in Math., vol. 434, Springer-Verlag, Berlin and New York, 1975. MR**0461121 (57:1106)**

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DOI:
https://doi.org/10.1090/S0025-5718-1977-0438664-4

Article copyright:
© Copyright 1977
American Mathematical Society