Finite element convergence for the time-dependent Joule heating problem with mixed boundary conditions

TL;DR

This paper establishes strong convergence results for finite element methods applied to the time-dependent Joule heating problem with mixed boundary conditions, accommodating adaptive meshes and irregular domain features.

Contribution

It provides the first convergence proof for finite element methods solving the Joule heating problem with mixed boundary conditions in three dimensions.

Findings

Proves strong convergence of finite element methods for the problem.

Establishes uniqueness and higher regularity of solutions.

Allows for adaptive mesh refinement due to variational formulation.

Abstract

We prove strong convergence for a large class of finite element methods for the time-dependent Joule heating problem in three spatial dimensions with mixed boundary conditions on Lipschitz domains. We consider conforming subspaces for the spatial discretization and the backward Euler scheme for the temporal discretization. Furthermore, we prove uniqueness and higher regularity of the solution on creased domains and additional regularity in the interior of the domain. Due to a variational formulation with a cut-off functional the convergence analysis does not require a discrete maximum principle, permitting approximation spaces suitable for adaptive mesh refinement, responding to the the difference in regularity within the domain.