Finite element convergence for the time-dependent Joule heating problem with mixed boundary conditions
Provides rigorous convergence guarantees for numerical simulation of Joule heating, a coupled electro-thermal problem, which is important for engineers and applied mathematicians working on such multiphysics systems.
The paper proves strong convergence for finite element methods applied to the time-dependent Joule heating problem in 3D with mixed boundary conditions, using backward Euler time-stepping and conforming spatial discretizations. It also establishes uniqueness and higher regularity on creased domains.
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.