ON THE IMPLEMENTATION OF MIXED METHODS
AS NONCONFORMING METHODS
FOR SECOND ORDER ELLIPTIC PROBLEMS

TODD ARBOGAST and ZHANGXIN CHEN

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 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.

Keywords: finite element, implementation, mixed method, equivalence,
nonconforming method, multigrid method