Finite Element Convergence for the Joule Heating Problem with Mixed Boundary Conditions

TL;DR

This paper proves the strong convergence of finite element methods applied to the stationary Joule heating problem with mixed boundary conditions, providing optimal regularity estimates and bounds for shape regular meshes in three dimensions.

Contribution

It establishes the first strong convergence results for finite element approximations of the Joule heating problem with mixed boundary conditions in 3D.

Findings

Proves strong convergence of finite element solutions.

Provides optimal regularity estimates on creased domains.

Derives a priori and a posteriori bounds for shape regular meshes.

Abstract

We prove strong convergence of conforming finite element approximations to the stationary Joule heating problem with mixed boundary conditions on Lipschitz domains in three spatial dimensions. We show optimal global regularity estimates on creased domains and prove a priori and a posteriori bounds for shape regular meshes.