An implicit-explicit BDF-Galerkin scheme of second order for the nonlinear thermistor problem

TL;DR

This paper introduces a second-order implicit-explicit BDF-Galerkin scheme for the nonlinear thermistor problem, providing stability and accuracy analysis along with numerical validation.

Contribution

It develops a novel second-order BDF-Galerkin scheme with unconditional stability and superconvergence for the nonlinear thermistor problem.

Findings

Unconditionally stable scheme demonstrated

Second-order convergence confirmed numerically

Effective handling of nonlinear coupling term

Abstract

This paper proposes and analyzes an implicit-explicit BDF-Galerkin scheme of second order for the time-dependent nonlinear thermistor problem. For this, we combine the second-order backward differentiation formula with special extrapolation terms for time discretization with standard finite elements for spatial discretization. Unconditionally superclose and superconvergent error estimates are established, relying on two key techniques. First, a time-discrete system is introduced to decompose the error function into its temporal and spatial components. Second, superclose error estimates between the numerical solution and the interpolation of the time-discrete solution are employed to effectively handle the nonlinear coupling term. Finally, we present numerical examples that validate the theoretical findings, demonstrating the unconditional stability and the second-order accuracy of the proposed method.