Finite element convergence analysis for the thermoviscoelastic Joule heating problem

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 firstorder in time. The theoretical results are verified by several numerical experiments in two and three dimensions.

The theta‐Galerkin finite element method for coupled systems resulting from microsensor thermistor problems

“…Theorem 3.1. Let (u, ) and (u h , h ) be the solutions of (5) to (7) and (19) to (21), respectively. Then there exists a constant C which does not depend on h such that…”

“…We write h − = ( h − Π h ) + (Π h − ) = Φ h + Λ and from Lemma 3.1, we have the bound of ||∇Λ|| L 12∕5 (Ω) . From (5) to (19), Φ h satisfies the following equation:…”

“…For the d ‐dimensional space ( d =2,3), Li et al presented the unconditionally optimal error estimates of a linearized Crank‐Nicolson Galerkin finite element method. Malqvist and Stillfjord presented the error estimate for finite element solutions of a thermoviscoelastic thermistor problem. However, to the best of our knowledge, there are very few study on numerical analysis of to for the general case.…”