New numerical studies for Darcy's problem coupled with the heat equation

In this article, we consider the heat equation coupled with Darcy's law by a nonlinear viscosity depending on the temperature. We recall two numerical schemes and introduce a new non-stabilized one, we show the existence and uniqueness of the solutions and we establish an a priori error estimates using the Brezzi-Rappaz-Raviart theorem. Numerical investigations are preformed and showed.