High order local approximations to derivatives in the finite element method
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- by Vidar Thomée PDF
- Math. Comp. 31 (1977), 652-660 Request permission
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 ${u_h}$ satisfies ${u_h} - u = O({h^r})$. Bramble and Schatz (Math. Comp., v. 31, 1977, pp. 94-111) have constructed, for elements satisfying certain uniformity conditions, a simple function ${K_h}$ such that ${K_h}\; \ast \;{u_h} - u = O({h^{2r - 2}})$ in the interior. Their result is generalized here to obtain similar superconvergent order interior approximations also for derivatives of u.References
- 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, DOI 10.1090/S0025-5718-1975-0398120-7
- 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 431744, DOI 10.1090/S0025-5718-1977-0431744-9
- 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 448926, DOI 10.1137/0714015
- 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, DOI 10.1007/BFb0068125
Additional Information
- © Copyright 1977 American Mathematical Society
- 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