Discrete duality finite volume scheme for a generalized Joule heating problem

In this paper we conceive and analyze a discrete duality finite volume (DDFV) scheme for the unsteady generalized thermistor problem, including a p-Laplacian for the diffusion and a Joule heating source. As in the continuous setting, the main difficulty in the design of the discrete model comes from the Joule heating term. To cope with this issue, the Joule heating term is replaced with an equivalent key formulation on which a fully implicit scheme is constructed. Introducing a tricky cut-off function in the proposed discretization, we are able to recover the energy estimates on the discrete temperature. Another feature of this approach is that we dispense with the discrete maximum principle on the approximate electric potential, which in essence poses restrictive constraints on the mesh shape. Then, the existence of discrete solution to the coupled scheme is established. Compactness estimates are also shown. Under general assumptions on the data and meshes, the convergence of the numerical scheme is addressed. Numerical results are finally presented to show the efficiency and accuracy of the proposed methodology as well as the behavior of the implemented nonlinear solver.