Two dimensional hierarchical mixed finite element approximations with enhanced primal variable accuracy

Abstract

The purpose of the present paper is to analyse different possibilities of choosing balanced pairs of approximation spaces for dual (flux) and primal (pressure) variables to be used in discrete versions of the mixed finite element method for affine two dimensional meshes. In all space configurations, the principle guiding their construction is the property that the divergence of the dual space and the primal approximation space should coincide, while keeping the same order of accuracy for the flux variable and varying the accuracy order of the primal variable. There is the classic case of BDMk spaces based on triangular meshes and polynomials of total degree k for the dual variable, and k − 1 for the primal variable, showing stable simulations with optimal convergence rates of orders k + 1 and k,respectively. Another case is related to RTk and BDF Mk+1 spaces for quadrilateral and triangular meshes, respectively. It gives identical approximation order k + 1 for both primal and dual variables, an improvement in accuracy obtained by increasing the degree of primal functions to k, and by enriching the dual space with some properly chosen internal shape functions of degree k + 1, while keeping degree k for the border fluxes. A new type of approximation is proposed by further incrementing the order of some internal flux functions to k + 2, and matching primal functions to k + 1 (higher than the border fluxes of degree k). Thus, higher convergence rate of order k + 2 is obtained for the primal variable. Using static condensation, the global condensed system to be solved in all the cases have same dimension (and structure), which is proportional to the space dimension of the border fluxes for each element geometry.