A finite element model for the time-dependent Joule heating problem

Abstract

We study a spatially semidiscrete and a completely discrete finite element model for a nonlinear system consisting of an elliptic and a parabolic partial differential equation describing the electric heating of a conducting body. We prove error bounds of optimal order under minimal regularity assumptions when the number of spatial variables d ≤ 3 d \leq 3 . We establish the existence of solutions with the required regularity over arbitrarily long intervals of time when d ≤ 2 d \leq 2 .