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

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.