On the Implementation of Mixed Methods as Nonconforming
Methods for Second Order Elliptic Problems

Todd Arbogast
Department of Computational and Applied Mathematics
Rice University, Houston, Texas 77251

Zhangxin Chen
Department of Mathematics and
Institute for Scientific Computation
Texas A&M University
College Station, TX 77843

Abstract

In this paper we show that mixed finite element methods for a fairly general second order elliptic problem with variable coefficients can be given a nonmixed formulation. (Lower order terms are treated, so our results apply also to parabolic equations.) We define an approximation method by incorporating some projection operators within a standard Galerkin method, which we call a projection finite element method. It is shown that for a given mixed method, if the projection method's finite element space Mh satisfies three conditions, then the two approximation methods are equivalent. These three conditions can be simplified for a single element in the case of mixed spaces possessing the usual vector projection operator. We then construct appropriate nonconforming spaces Mh for the known triangular and rectangular elements. The lowest-order Raviart-Thomas mixed solution on rectangular finite elements in R2 and R3, on simplices, or on prisms, is then implemented as a nonconforming method modified in a simple and computationally trivial manner. This new nonconforming solution is actually equivalent to a postprocessed version of the mixed solution. A rearrangement of the computation of the mixed method solution through this equivalence allows us to design simple and optimal order multigrid methods for the solution of the linear system.


Contributed January 20, 1995.