Remarks on Mixed Finite Element Methods for Problems with Rough Coefficients

Abstract

This paper considers the finite element approximation of elliptic boundary value problems in divergence form with rough coefficients. The so- lution 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. This paper is concerned with the finite element approximation of elliptic boundary value problems in divergence form with rough coefficients. The solu- tions of such problems will in general be rough, and it is well known that the usual (Ritz or displacement) finite element method based on piecewise linear approximating functions is 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. Mixed variational principles arise naturally in the mathematical formulation of many physical problems. For example, the laws of linear elasticity may be described in terms of a displacement variational formulation, involving only displacements, or in terms of a mixed variational formulation, involving both stresses and displace- ments, whose equations express the stress-strain relation and the balance of forces. In purely mathematical terms, one can obtain a mixed formulation from a displacement formulation by introducing new variables for some of the derivatives or certain linear combinations of derivatives of the unknown func- tion. A mixed approximation method is obtained by basing a finite element