A Second Order Unconditionally Convergent Finite Element Method for the Thermal Equation with Joule Heating Problem

Authors

  • Xiaonian Long College of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450045, China
  • Qianqian Ding School of Mathematics, Shandong University, Jinan 250100, China

DOI:

https://doi.org/10.4208/jcm.2010-m2020-0145

Keywords:

Thermal equation, Joule heating, Finite element method, Unconditional convergence, Second order backward difference formula, Optimal $L^2$-estimate.

Abstract

In this paper, we study the finite element approximation for nonlinear thermal equation. Because the nonlinearity of the equation, our theoretical analysis is based on the error of temporal and spatial discretization. We consider a fully discrete second order backward difference formula based on a finite element method to approximate the temperature and electric potential, and establish optimal $L^2$ error estimates for the fully discrete finite element solution without any restriction on the time-step size. The discrete solution is bounded in infinite norm. Finally, several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

Published

2022-10-06

Issue

Section

Articles