### 2010-15

by Braack, M.; Schieweck, F..

**Series:** 2010-15, Preprints

- MSC:
- 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

**Abstract:**

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.

**Keywords:**

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