On the implementation of mixed methods as nonconforming methods for second-order elliptic problems

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 ${\mathbb {R}^2}$ and ${\mathbb {R}^3}$, 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.