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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 $ {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 [Enhancements On Off] (What's this?)

  • [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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Article copyright: © Copyright 1977 American Mathematical Society

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