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

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. We establish the existence of solutions with the required regularity over arbitrarily long intervals of time when d < 2 .

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