Numerical analysis of a nonlocal parabolic problem resulting from thermistor problem

TL;DR

This paper conducts a numerical analysis of finite element methods for a nonlocal parabolic thermistor problem, providing error estimates and algorithms for fully discrete schemes.

Contribution

It introduces an analysis framework for nonlocal parabolic equations from thermistor problems, including error estimates and solution algorithms for various discretizations.

Findings

Optimal order error estimates achieved

Analysis covers backward Euler and Crank-Nicolson schemes

Proposed simple algorithm for fully discrete problem

Abstract

We analyze the spatially semidiscrete piecewise linear finite element method for a nonlocal parabolic equation resulting from thermistor problem. Our approach is based on the properties of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite element method. We assume minimal regularity of the exact solution that yields optimal order error estimate. The full discrete backward Euler method and the Crank-Nicolson-Galerkin scheme are also considered. Finally, a simple algorithm for solving the fully discrete problem is proposed.