Mixed methods using standard conforming finite elements

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 Qk×(Qk)dQk×(Qk)d on rectangles or Pk×(Pk)dPk×(Pk)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 Qk×(Qk)dQk×(Qk)d with any k⩾1k⩾1, or for P1×(P1)dP1×(P1)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 Q1×(Q1)2Q1×(Q1)2 and P1×(P1)2P1×(P1)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.