Derivative superconvergent points in finite element solutions of harmonic functions-- A theoretical justification

Author:
Zhimin Zhang

Journal:
Math. Comp. **71** (2002), 1421-1430

MSC (2000):
Primary 65N30

DOI:
https://doi.org/10.1090/S0025-5718-01-01398-9

Published electronically:
December 5, 2001

MathSciNet review:
1933038

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Abstract | References | Similar Articles | Additional Information

Abstract: Finite element derivative superconvergent points for harmonic functions under local rectangular mesh are investigated. All superconvergent points for the finite element space of any order that is contained in the tensor-product space and contains the intermediate family can be predicted. In the case of the serendipity family, results are given for finite element spaces of order below 6. The results justify the computer findings of Babuska, et al.

**1.**I. Babuska and T. Strouboulis,*The Finite Element Method and its Reliability*, Oxford University Press, London, 2001.**2.**I. Babuska, T. Strouboulis, C.S. Upadhyay, and S.K. Gangaraj,*Computer-based proof of the existence of superconvergence points in the finite element method; superconvergence of the derivatives in finite element solutions of Laplace's, Poisson's, and the elasticity equations*, Numer. Methods Partial Differential Equations**12**(1996), 347-392. MR**97c:65160****3.**M. Krízek, P. Neittaanmäki, and R. Stenberg (Eds.),*Finite element methods. Superconvergence, post-processing, and a posteriori estimates*, Lecture Notes in Pure and Applied Mathematics Series, Vol. 196, Marcel Dekker, New York, 1997. MR**98i:65003****4.**N.N. Lebedev,*Special functions and their applications*, Dover, New York, 1972. MR**50:2568****5.**A.H. Schatz, I.H. Sloan, and L.B. Wahlbin,*Superconvergence in finite element methods and meshes which are symmetric with respect to a point*, SIAM J. Numer. Anal.**33(2)**(1996), 505-521. MR**98f:65112****6.**L.B. Wahlbin,*Superconvergence in Galerkin finite element methods*, Lecture Notes in Mathematics, Vol. 1605, Springer, Berlin, 1995. MR**98j:65083****7.**Zhimin Zhang,*Derivative superconvergent points in finite element solutions of Poisson's equation for the serendipity and intermediate families--A theoretical justification*, Math. Comp.**67**(1998), 541-552. MR**98i:65104**

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Additional Information

**Zhimin Zhang**

Affiliation:
Department of Mathematics, Wayne State University, Detroit, Michigan 48202

Email:
zzhang@math.wayne.edu

DOI:
https://doi.org/10.1090/S0025-5718-01-01398-9

Keywords:
Superconvergence,
finite element,
harmonic function

Received by editor(s):
November 21, 2000

Published electronically:
December 5, 2001

Article copyright:
© Copyright 2001
American Mathematical Society