Optimal error estimates and recovery technique of a mixed finite element method for nonlinear thermistor equations

This paper is concerned with optimal error estimates and recovery technique of a classical mixed finite element method for the thermistor problem, which is governed by a parabolic/elliptic system with strong nonlinearity and coupling. The method is based on a popular combination of the lowest-order Raviart–Thomas mixed approximation for the electric potential/field (φ, θ) and the linear Lagrange approximation for the temperature u. A common question is how the first-order approximation influences the accuracy of the second-order approximation to the temperature in such a strongly coupled system, while previous work only showed the first-order accuracy O(h) for all three components in a traditional way. In this paper, we prove that the method produces the optimal second-order accuracy O(h²) for u in the spatial direction, although the accuracy for the potential/field is in the order of O(h). And more importantly, we propose a simple one-step recovery technique to obtain a new numerical electric potential/field of second-order accuracy. The analysis presented in this paper relies on an H⁻¹-norm estimate of the mixed finite element methods and analysis on a nonclassical elliptic map. We provide numerical experiments in both two- and three-dimensional spaces to confirm our theoretical analyses.