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Huadong Gao, Weiwei Sun, Chengda Wu, Optimal error estimates and recovery technique of a mixed finite element method for nonlinear thermistor equations, IMA Journal of Numerical Analysis, Volume 41, Issue 4, October 2021, Pages 3175–3200, https://doi.org/10.1093/imanum/draa063
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Abstract
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 |$(\phi , \boldsymbol{\theta })$| 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 |$O(h^2)$| 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.
