High order local approximations to derivatives in the finite element method
Math. Comp. 31 (1977), 652-660
Primary 65D25; Secondary 65L10
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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.
H. Bramble, Joachim
A. Nitsche, and Alfred
H. Schatz, Maximum-norm interior estimates for
Ritz-Galerkin methods, Math. Comput. 29 (1975), 677–688.
0398120 (53 #1975), http://dx.doi.org/10.1090/S0025-5718-1975-0398120-7
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
(55 #4739), http://dx.doi.org/10.1090/S0025-5718-1977-0431744-9
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
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, 1975. MR 0461121
- 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)
- 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)
- 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)
- 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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