Two-Dimensional H(div)-Conforming Finite Element Spaces with hp-Adaptivity

Abstract

The purpose of the paper is to analyse the effect of hp mesh adaptation when discretized versions of finite element mixed formulations are applied to elliptic problems with singular solutions. Two stable configurations of approximation spaces, based on affine triangular and quadrilateral meshes, are considered for primal and dual (flux) variables. When computing sufficiently smooth solutions using regular meshes, the first configuration gives optimal convergence rates of identical approximation orders for both variables, as well as for the divergence of the flux. For the second configuration, higher convergence rates are obtained for the primal variable. Furthermore, after static condensation is applied, the condensed systems to be solved have the same dimension in both configuration cases, which is proportional to their border flux dimensions. A test problem with a steep interior layer is simulated, and the results demonstrate exponential rates of convergence. Comparison of the results obtained with H1-conforming formulation are also presented.