Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations

TL;DR

This paper provides optimal error estimates for semi-implicit Galerkin finite element methods applied to nonlinear parabolic equations, notably Joule heating, without requiring time-step restrictions.

Contribution

It introduces a new error splitting technique that allows for optimal error estimates without time-step restrictions, applicable to various nonlinear parabolic systems.

Findings

Optimal error estimates in L2 and H1 norms for semi-implicit Euler scheme

No time-step restriction needed for convergence

Applicable to general nonlinear parabolic systems

Abstract

This paper is concerned with the time-step condition of commonly-used linearized semi-implicit schemes for nonlinear parabolic PDEs with Galerkin finite element approximations. In particular, we study the time-dependent nonlinear Joule heating equations. We present optimal error estimates of the semi-implicit Euler scheme in both the $L^2$ norm and the $H^1$ norm without any time-step restriction. Theoretical analysis is based on a new splitting of the error and precise analysis of a corresponding time-discrete system. The method used in this paper can be applied to more general nonlinear parabolic systems and many other linearized (semi)-implicit time discretizations for which previous works often require certain restriction on the time-step size $\tau$.