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. In our analysis, 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.