Methods for Second Order Elliptic Problems

Department of Computational and Applied Mathematics

Rice University, Houston, Texas 77251

Department of Mathematics and

Institute for Scientific Computation

Texas A&M University

College Station, TX 77843

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
M_{h} 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 M_{h} for the known
triangular and rectangular elements. The lowest-order Raviart-Thomas mixed
solution on rectangular finite elements in R^{2} and R^{3}, 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.