Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Remarks on mixed finite element methods for problems with rough coefficients

Authors: Richard S. Falk and John E. Osborn
Journal: Math. Comp. 62 (1994), 1-19
MSC: Primary 65N30
MathSciNet review: 1203735
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper considers the finite element approximation of elliptic boundary value problems in divergence form with rough coefficients. The solution of such problems will, in general, be rough, and it is well known that the usual (Ritz or displacement) finite element method will be inaccurate in general. The purpose of the paper is to help clarify the issue of whether the use of mixed variational principles leads to finite element schemes, i.e., to mixed methods, that are more accurate than the Ritz or displacement method for such problems. For one-dimensional problems, it is well known that certain mixed methods are more accurate and robust than the Ritz method for problems with rough coefficients. Our results for two-dimensional problems are mostly of a negative character. Through an examination of examples, we show that certain standard mixed methods fail to provide accurate approximations for problems with rough coefficients except in some special situations.

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

  • [1] I. Babuška, G. Caloz, and J. E. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM J. Numer. Anal. (to appear). MR 1286212 (95g:65146)
  • [2] I. Babuška and J. E. Osborn, Generalized finite element methods: their performance and their relation to mixed methods, SIAM J. Numer. Anal. 20 (1983), 510-536. MR 701094 (84h:65076)
  • [3] -, Finite element methods for the solution of problems with rough data, Singularities and Constructive Methods for Their Treatment (P. Grisvard, W. Wendland, and J. R. Whiteman, eds.), Lecture Notes in Math., vol. 1121, Springer-Verlag, Berlin and New York, 1985, pp. 1-18.
  • [4] S. N. Bernstein, Sur la généralization du problème de Dirichlet, Math. Ann. 62 (1906), 253-272; ibid. 69 (1910), 82-136. MR 1511579
  • [5] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer-Verlag, New York, 1991. MR 1115205 (92d:65187)
  • [6] R. Falk and J. Osborn, Error estimates for mixed methods, RAIRO 14 (1980), 249-277. MR 592753 (82j:65076)
  • [7] P. Grisvard, Elliptic problems in non-smooth domains, Pitman Advanced Publishing Program, Boston-London-Melbourne, 1985. MR 775683 (86m:35044)
  • [8] O. Ladyzhenskaia and N. Ural'tseva, Linear and quasilinear elliptic equations, Academic Press, New York, 1968. MR 0244627 (39:5941)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N30

Retrieve articles in all journals with MSC: 65N30

Additional Information

Keywords: Finite elements, mixed methods
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society