Negative norm estimates and superconvergence in Galerkin methods for parabolic problems

Thomée, Vidar

doi:10.1090/S0025-5718-1980-0551292-5

Negative norm estimates and superconvergence in Galerkin methods for parabolic problems
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Math. Comp. 34 (1980), 93-113 Request permission

Abstract:

Negative norm error estimates for semidiscrete Galerkin-finite element methods for parabolic problems are derived from known such estimates for elliptic problems and applied to prove superconvergence of certain procedures for evaluating point values of the exact solution and its derivatives.

References

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
Jim Douglas Jr., Todd Dupont, and Mary F. Wheeler, A quasi-projection analysis of Galerkin methods for parabolic and hyperbolic equations, Math. Comp. 32 (1978), no. 142, 345–362. MR 495012, DOI 10.1090/S0025-5718-1978-0495012-2
Vidar Thomée, High order local approximations to derivatives in the finite element method, Math. Comp. 31 (1977), no. 139, 652–660. MR 438664, DOI 10.1090/S0025-5718-1977-0438664-4
Mary Fanett Wheeler, A priori $L_{2}$ error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal. 10 (1973), 723–759. MR 351124, DOI 10.1137/0710062
Mary Fanett Wheeler, $L_{\infty }$ estimates of optimal orders for Galerkin methods for one-dimensional second order parabolic and hyperbolic equations, SIAM J. Numer. Anal. 10 (1973), 908–913. MR 343658, DOI 10.1137/0710076