Equal-order finite elements with local projection stabilization for the Darcy–Brinkman equations

For the Darcy–Brinkman equations, which model porous media flow, we present an equal-order H1-conforming finite element method for approximating velocity and pressure based on a local projection stabilization technique. The method is stable and accurate uniformly with respect to the coefficients of the viscosity and the zeroth order term in the momentum equation. We prove a priori error estimates in a mesh-dependent norm as well as in the L2-norm for velocity and pressure. In particular, we obtain optimal order of convergence in L2 for the pressure in the Darcy case with vanishing viscosity and for the velocity in the general case with a positive viscosity coefficient. Numerical results for different values of the coefficients in the Darcy–Brinkman model are presented which confirm the theoretical results and indicate nearly optimal order also in cases which are not covered by the theory.