A Finite Element Model for the Time-Dependent Joule Heating Problem

Abstract

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

Finite element convergence analysis for the thermoviscoelastic Joule heating problem

“…L 1 (0, T ; L 2 ) ≤ C The assumptions on θ and φ are essentially the same as in the non-deformable situation given in [23], while the assumptions on u and f are new. We note that for the non-deformable case, the existence of solutions with similar regularity properties was shown in [11] when d ≤ 2, with weak requirements on the initial values. In the general elliptic/parabolic case, the absence of reentrant corners in the convex domain makes such regularity plausible, see e.g.…”

“…In order to do this, we will generalize the analysis of [23] (cf. also [11]) for the case with no deformation. This consists of first showing that the time-discrete approximations are O(k)-close to the solutions of the continuous system, and also proving that these approximations exhibit a certain regularity.…”

“…Similarly, the quasi-uniformity of the mesh guarantees that the standard inverse inequalities are satisfied. These are needed to handle the nonlinear potential term in (1), see [11,23].…”

“…Many authors have analyzed methods for similar problems. For example, [12] considers the quasi-static version where the ü-term is ignored, [1], [11] and [22] considers the non-deformable case, [13,14] treat the purely thermoviscoelastic case (no φ) with nonlinear constituent law, etc. Additionally, in the deformable case a common theme seems to be suboptimal convergence orders, i.e.…”

“…The central idea of our proof is to bound the errors in φ and u in terms of the error in θ, in the spirit of [11] and [23]. The latter error then fulfills an equation similar to (1), to which we may apply a Grönwall inequality after properly handling the quadratic potential term.…”