Unconditionally optimal error estimates of a Crank--Nicolson Galerkin method for the nonlinear thermistor equations

TL;DR

This paper establishes unconditionally optimal error estimates for a Crank--Nicolson Galerkin finite element method applied to nonlinear thermistor equations, providing rigorous analysis and numerical validation.

Contribution

It provides the first unconditionally optimal error analysis for a linearized Crank--Nicolson Galerkin method on nonlinear thermistor equations, including regularity and error estimates.

Findings

Error estimates are unconditionally optimal.

Numerical results confirm theoretical analysis.

Method demonstrates high efficiency and accuracy.

Abstract

This paper focuses on unconditionally optimal error analysis of an uncoupled and linearized Crank--Nicolson Galerkin finite element method for the time-dependent nonlinear thermistor equations in $d$-dimensional space, $d=2,3$. We split the error function into two parts, one from the spatial discretization and one from the temporal discretization, by introducing a corresponding time-discrete (elliptic) system. We present a rigorous analysis for the regularity of the solution of the time-discrete system and error estimates of the time discretization. With these estimates and the proved regularity, optimal error estimates of the fully discrete Crank--Nicolson Galerkin method are obtained unconditionally. Numerical results confirm our analysis and show the efficiency of the method.