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 the Institute
   for Scientific Computation, Texas A&M University,
            College Station, Texas 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 $M_h$
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 $\Re^2$ and $\Re^3$, 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.