Numerical discretizations of a convective Brinkman-Forchheimer model under singular forcing

Abstract

In two-dimensional Lipschitz domains, we analyze a Brinkman--Darcy--Forchheimer problem on the weighted spaces ${\mathbf{H}}_0^1(\omega ,\Omega )\times L^2(\omega ,\Omega )/\mathbb{R}$, where $\omega$ belongs to the Muckenhoupt class $A_2$. Under a suitable smallness assumption, we prove the existence and uniqueness of a solution. We propose a finite element method and obtain a quasi-best approximation result in the energy norm \emph{à la Céa} under the assumption that $\Omega$ is convex. We also develop an a posteriori error estimator and study its reliability and efficiency properties. Finally, we develop an adaptive method that yields optimal experimental convergence rates for the numerical examples we perform.