Numerical discretization of a Darcy-Forchheimer problem coupled with a singular heat equation

In Lipschitz domains, we study a Darcy-Forchheimer problem coupled with a singular heat equation by a nonlinear forcing term depending on the temperature. By singular we mean that the heat source corresponds to a Dirac measure. We establish the existence of solutions for a model that allows a diffusion coefficient in the heat equation depending on the temperature. For such a model, we also propose a finite element discretization scheme and provide an a priori convergence analysis. In the case that the aforementioned diffusion coefficient is constant, we devise an a posteriori error estimator and investigate reliability and efficiency properties. We conclude by devising an adaptive loop based on the proposed error estimator and presenting numerical experiments.