Two dimensional mixed finite element approximations for elliptic problems with enhanced accuracy for the potential and flux divergence

Abstract

The purpose of the present paper is to analyse two new different possibilities of choosing balanced pairs of approximation spaces for dual (flux) and primal (potential) variables, one for triangles and the other one for quadrilateral elements, to be used in discrete versions of the mixed finite element method for elliptic problems. They can be interpreted as enriched versions of BDFMk+1 spaces based on triangles, and of RTk spaces for quadrilateral elements. The new flux approximations are incremented with properly chosen internal shape functions (with vanishing normal components on the edges) of degree k+2, and matching primal functions of degree k+1 (higher than the border fluxes, which are kept of degree k). In all these cases, the divergence of the flux space coincide with the primal approximation space on the master element, producing stable simulations. Using static condensation, the global condensed system to be solved in the enriched cases has same dimension (and structure) of the original ones, which is proportional to the space dimension of the border fluxes for each element geometry. Measuring the errors withL2-norms, the enriched space configurations give higher convergence rate of order k+2 for the primal variable, while keeping the order k+1 for the flux. For affine meshes, the divergence errors have the same improved accuracy rate as for the error in the primal variable. For quadrilateral non-affine meshes, for instance trapezoidal elements, the divergence error has order k+1, one unit more than the order k occurring for RTk spaces on this kind of deformed meshes. This fact also holds for ABFk elements, but for them the potential order of accuracy does not improve, keeping order k+1.