T. Arbogast et Zx. Chen, ON THE IMPLEMENTATION OF MIXED METHODS AS NONCONFORMING METHODS FOR 2ND-ORDER ELLIPTIC PROBLEMS, Mathematics of computation, 64(211), 1995, pp. 943-972
In this paper we show that mixed finite element methods for a fairly g
eneral 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 G
alerkin method, which we call a projection finite element method. It i
s shown that for a given mixed method, if the projection method's fini
te element space M(h) satisfies three conditions, then the two approxi
mation methods are equivalent. These three conditions can be simplifie
d for a single element in the case of mixed spaces possessing the usua
l vector projection operator. We then construct appropriate nonconform
ing spaces M(h) for the known triangular and rectangular elements. The
lowest-order Raviart-Thomas mixed solution on rectangular finite elem
ents in R(2) and R(3), on simplices, or on prisms, is then implemented
as a nonconforming method modified in a simple and computationally tr
ivial manner. This new nonconforming solution is actually equivalent t
o a postprocessed version of the mixed solution. A rearrangement of th
e computation of the mixed method solution through this equivalence al
lows us to design simple and optimal-order multigrid methods for the s
olution of the linear system.