On the Implementation of Mixed Methods

In this paper we show that mixed nite element methods for a fairly general second order elliptic problem with variable coeecients can be given a non-mixed formulation. (Lower order terms are treated, so our results apply also to parabolic equations.) We deene an approximation method by incorporating some projection operators within a standard Galerkin method, which we call a projection nite element method. It is shown that for a given mixed method, if the projection method's nite element space M h satisses three conditions, then the two approximation methods are equivalent. These three conditions can be simpliied 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 nite elements in IR 2 and IR 3 , on simplices, or on prisms, is then implemented as a nonconforming method modiied 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.