A Second Order Unconditionally Convergent Finite Element Method for the Thermal Equation with Joule Heating Problem (Journal of Computational Mathematics, 2022)
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.
Keywords: Thermal equation; Joule heating; Finite element method; Unconditional convergence; Second order backward difference formula; Optimal $L^2$-estimate.