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

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]**James H. Bramble, Joachim A. Nitsche, and Alfred H. Schatz,*Maximum-norm interior estimates for Ritz-Galerkin methods*, Math. Comput.**29**(1975), 677–688. MR**0398120**, 10.1090/S0025-5718-1975-0398120-7**[2]**J. H. Bramble and A. H. Schatz,*Higher order local accuracy by averaging in the finite element method*, Math. Comp.**31**(1977), no. 137, 94–111. MR**0431744**, 10.1090/S0025-5718-1977-0431744-9**[3]**J. H. Bramble, A. H. Schatz, V. Thomée, and L. B. Wahlbin,*Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations*, SIAM J. Numer. Anal.**14**(1977), no. 2, 218–241. MR**0448926****[4]**Philip Brenner, Vidar Thomée, and Lars B. Wahlbin,*Besov spaces and applications to difference methods for initial value problems*, Lecture Notes in Mathematics, Vol. 434, Springer-Verlag, Berlin-New York, 1975. MR**0461121**

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

Article copyright:
© Copyright 1977
American Mathematical Society