Finite element convergence analysis for the thermoviscoelastic Joule heating problem

TL;DR

This paper presents a finite element method for modeling thermoviscoelastic Joule heating, proving optimal convergence rates and validating results through numerical experiments in multiple dimensions.

Contribution

It introduces a fully discretized finite element approach with proven optimal convergence for the thermoviscoelastic Joule heating problem.

Findings

Optimal second-order spatial convergence

First-order temporal convergence

Numerical validation in 2D and 3D

Abstract

We consider a system of equations that model the temperature, electric potential and deformation of a thermoviscoelastic body. A typical application is a thermistor; an electrical component that can be used e.g. as a surge protector, temperature sensor or for very precise positioning. We introduce a full discretization based on standard finite elements in space and a semi-implicit Euler-type method in time. For this method we prove optimal convergence orders, i.e. second-order in space and first-order in time. The theoretical results are verified by several numerical experiments in two and three dimensions.