Darcy’s problem coupled with the heat equation under singular forcing: analysis and discretization

We study the existence of solutions for Darcy’s problem coupled with the heat equation under singular forcing; the right-hand side of the heat equation corresponds to a Dirac measure. The model studied involves thermal diffusion and viscosity depending on the temperature. We propose a finite element solution technique and analyze its convergence properties. In the case where thermal diffusion is independent of temperature, we propose an a posteriori error estimator and study its reliability and efficiency properties. We illustrate the theory with numerical examples.