Hierarchical high order finite element spaces in [formula omitted] for a stabilized mixed formulation of Darcy problem

Abstract

The classical dual mixed finite element method for flow simulations is based on H(div,Ω) conforming approximation spaces for the flux, which guarantees continuous normal components on element interfaces, and discontinuous approximations in L2(Ω) for the pressure. However, stability and convergence can only be obtained for compatible approximation spaces. Stabilized finite element methods may provide an alternative stable procedure to avoid this kind of delicate balance. The main purpose of this paper is to present a high-order finite element methodology to solve the Darcy problem based on the combination of an unconditionally stable mixed finite element method with a hierarchical methodology for the construction of finite dimensional subspaces of H(div,Ω) and H1(Ω). The chosen stabilized method is free of mesh dependent stabilization parameters and allows for the use of different high order finite element approximations for the flux and the pressure variables, without requiring any compatibility constraint, as required in mixed methods for these problems. Convergence studies are presented comparing the numerical solutions obtained for different approximation orders on quadrilateral elements with the ones given by classical mixed formulation with Raviart–Thomas elements.