A weak Galerkin pseudostress-based mixed finite element method on polygonal meshes: application to the Brinkman problem appearing in porous media

Abstract

In this paper, we extend the utilization of pseudostress-based formulation, recently employed for solving diverse linear and nonlinear problems in continuum mechanics via mixed finite element methods, to the weak Galerkin method (WG) framework and its respective applications. More precisely, we propose and analyze a mixed weak Galerkin method for a pseudostress formulation of the two-dimensional Brinkman equations with Dirichlet boundary conditions, then compute the velocity and pressure via postprocessing formulae. We begin by recalling the corresponding continuous variational formulation and a summary of the main WG method, including the weak divergence operator and the discrete space, which are needed for our approach. In particular, in order to define the weak discrete bilinear form, whose continuous version involves the classical divergence operator, we propose the weak divergence operator as a well-known alternative for the classical divergence operator in a suitable discrete subspace. Next, we show that the discrete bilinear form satisfies the hypotheses required by the Lax–Milgram lemma. In this way, we prove the well-posedness of the weak Galerkin scheme and derive a priori error estimates for the numerical pseudostress, velocity, and pressure. Finally, several numerical results confirming the theoretical rates of convergence and illustrating the good performance of the method are presented. The results in this work are fundamental and can be extended into more relevant models.