Mixed methods using standard conforming finite elements

Abstract

We investigate the mixed finite element method (MFEM) for solving a second order elliptic problem with a lowest order term, as might arise in the simulation of single-phase flow in porous media. We find that traditional mixed finite element spaces are not necessary when a positive lowest order (i.e., reaction) term is present. Hence, we propose to use standard conforming finite elements Q k × ( Q k ) d on rectangles or P k × ( P k ) d on simplices to solve for both the pressure and velocity field in d dimensions. The price we pay is that we have only sub-optimal order error estimates. With a delicate superconvergence analysis, we find some improvement for the simplest pair Q k × ( Q k ) d with any k ⩾ 1 , or for P 1 × ( P 1 ) d , when the mesh is uniform and the solution has one extra order of regularity. We also prove similar results for both parabolic and second order hyperbolic problems. Numerical results using Q 1 × ( Q 1 ) 2 and P 1 × ( P 1 ) 2 are presented in support of our analysis. These observations allow us to simplify the implementation of the MFEM, especially for higher order approximations, as might arise in an hp-adaptive procedure.