A posteriori error estimates for Darcy's problem coupled with the heat equation

This work derives a posteriori error estimates, in two and three dimensions, for the heat equation coupled with Darcy’s law by a nonlinear viscosity depending on the temperature. We introduce two variational formulations and discretize them by finite element methods. We prove optimal a posteriori errors with two types of computable error indicators. The first one is linked to the linearization and the second one to the discretization. Then we prove upper and lower error bounds under regularity assumptions on the solutions. Finally, numerical computations are performed to show the effectiveness of the error indicators.