An a posteriori error estimator for a new stabilized formulation of the Brinkman problem

Abstract

We present in this work an a posteriori error estimator for a porous media flow problem that follows the Brinkman model. First, we introduce the pseudostress as an auxiliary unknown, which let us to eliminate the pressure and thus derive a dual-mixed formulation in velocity-pseudostress. Next, in order to circumvent an inf-sup condition for the unique solvability, we stabilize the scheme by adding some appropriate least squares terms. The existence and uniqueness of solution are guaranteed and we derive an a posteriori error estimator based on the Ritz projection of the error, which is reliable and efficient up to high order terms. Finally, we report one numerical example confirming the good properties of the estimator.