Can a finite element method

perform arbitrarily badly?

Authors:
Ivo Babuska and John E. Osborn

Journal:
Math. Comp. **69** (2000), 443-462

MSC (1991):
Primary :, 65N15, 65N30

DOI:
https://doi.org/10.1090/S0025-5718-99-01085-6

Published electronically:
February 24, 1999

MathSciNet review:
1648351

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we construct elliptic boundary value problems whose standard finite element approximations converge arbitrarily slowly in the energy norm, and show that adaptive procedures cannot improve this slow convergence. We also show that the -norm and the nodal point errors converge arbitrarily slowly. With the -norm two cases need to be distinguished, and the usual duality principle does not characterize the error completely. The constructed elliptic problems are one dimensional.

**1.**A. Aubin,*Behavior of the error of the approximate solution of boundary value problems for linear elliptic operators by Galerkin's and finite difference methods*, Ann. Scoula Norm. Sup. Pisa**21**(1967), 599-637. MR**38:1391****2.**I. Babu\v{s}ka G. Caloz, and J. Osborn,*Special finite element methods for a class of second order boundary value problems with rough coefficients*, SIAM J. Numer. Anal.**31**(1994), 945-981. MR**95g:65146****3.**I. Babu\v{s}ka and J.M. Melenk,*The partition of unity method*, Int. J. Numer. Meth. Eng.**40**(1997), 727-758. MR**97j:73071****4.**I. Babu\v{s}ka and A. Miller,*A feedback finite element method with a posteriori error estimation: Part I. The finite element method and some basic properties of the a posteriori error estimator*, Comput. Methods Appl. Mech. Engrg.**61**(1987), 1-40. MR**88d:73036****5.**I. Babu\v{s}ka and J. E. Osborn,*Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems*, Math. Comp.**52**(1989), 275-297. MR**89k:65132****6.**I. Babu\v{s}ka and M. Suri,*The p and h-p versions of the finite element method, basic principles and properties*, SIAM Review**36**(1994), 578-632. MR**96d:65184****7.**I. Babu\v{s}ka and M. Vogelius,*Feedback and adaptive finite element solution of one dimensional boundary value problem*, Numer. Math.**44**(1984), 75-102. MR**85k:65070****8.**S.N. Bernstein,*On the inverse problem of best approximation of continuous functions*, Sochineniya**II**(1938), 292-294.**9.**R. DeVore and F. Richards,*Saturation and inverse theorems for spline approximation*, in Spline Functions and Approximation Theory, A. Meir and A. Sharma, eds., Birkhäuser Verlag, 1973, pp. 73-82. MR**51:8698****10.**J. R. Higgins,*Completeness and Basis Properties of Sets of Special Functions*, Cambridge University Press, New York, 1977. MR**58:17240****11.**J. Nitsche,*Ein Kriterium für die Quasi-Optimalität des Ritzschen Verfahren*, Numer. Math.**11**(1968), 346-348. MR**38:1823****12.**L.A. Oganesyan, and L.A. Rukhovets,*Investigation of the convergence rate of variational-difference schemes for second order elliptic equations in a two dimensional domain with a smooth boundary*, (English translation) U.S.S.R. Comput. Math. Phys.**9**(1969), no. 5, 158-183. MR**45:4665****13.**P. Oswald, Personal Communication.**14.**A. Pinkus,*-Widths in Approximation Theory*, Springer-Verlag, Berlin, 1985. MR**86k:41001****15.**A Schatz and J. Wang,*Some new error estimates for Ritz-Galerkin methods with minimal regularity assumptions*, Math. Comp.**65**(1996), 19-27. MR**96d:65190****16.**A.F. Timan,*Theory of approximation of functions of a real variable*, Macmillian, New York, 1963. MR**33:465**

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Additional Information

**Ivo Babuska**

Affiliation:
Texas Institute for Computational and Applied Mathematics, University of Texas at Austin, Austin, TX 78712

**John E. Osborn**

Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742

Email:
jeo@math.umd.edu

DOI:
https://doi.org/10.1090/S0025-5718-99-01085-6

Keywords:
Finite element methods,
convergence,
adaptivity,
rough coefficients

Received by editor(s):
May 5, 1998

Published electronically:
February 24, 1999

Additional Notes:
The first author was supported in part by NSF Grant #DMS-95-01841.

Article copyright:
© Copyright 2000
American Mathematical Society