A finite volume method for a nonlocal thermistor problem

In this work, we consider a more general version of the nonlocal thermistor problem, which describes the temperature diffusion produced when an electric current passes through a material. We investigate the doubly nonlinear problem where the nonlocal term is present on the right-hand side of the equation that describes the temperature evolution. Specifically, we employ topological degree theory to establish the existence of a solution to the considered problem. Furthermore, we separately address the uniqueness of the obtained solution. Additionally, we establish a priori estimates to demonstrate the convergence of a developed finite volume scheme used for the discretization of the continuous parabolic problem. Finally, to numerically simulate the proposed finite volume scheme, we use the Picard-type iteration process for the fully implicit scheme and approximate the nonlocal term represented by the integral with Simpson's rule to validate the efficiency and robustness of the proposed scheme.