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

by Braack, M.; Schieweck, F..

Series: 2010-15, Preprints

65N30 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods
65N12 Stability and convergence of numerical methods
65N15 Error bounds
76D07 Stokes and related (Oseen, etc.) flows

For the Darcy-Brinkman equations, which model porous media flow, we present an equal-order $H^1$-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 $L^2$-norm for velocity and pressure. In particular, we obtain optimal order of convergence in $L^2$ 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.

Porous media flow, Darcy-Brinkman equations, Stokes, equal-order finite elements, local projection stabilization