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.

中文翻译：

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非线性热敏电阻方程的混合有限元方法的最佳误差估计和恢复技术

本文涉及热敏电阻问题的经典混合有限元方法的最佳误差估计和恢复技术，该方法由具有强非线性和耦合特性的抛物线/椭圆系统控制。该方法基于电势/场| $（\ phi，\ boldsymbol {\ theta}）$ |的最低阶Raviart–Thomas混合逼近的流行组合。以及温度| $ u $ |⁠的线性拉格朗日逼近。一个常见的问题是在这种强耦合系统中，一阶逼近如何影响温度的二阶逼近精度，而先前的工作仅显示了一阶精度| $ O（h）$ |。以传统方式处理所有三个组件。在本文中，我们证明了该方法产生了最佳的二阶精度| $ O（h ^ 2）$ | 为| $ u $ | 在空间方向上，尽管势/场的精度约为| $ O（h）$ |⁠。更重要的是，我们提出了一种简单的单步恢复技术，以获得具有二阶精度的新数值电势/场。本文提供的分析依赖于| $ H ^ {-1} $ | -范数估计的混合有限元方法和非经典椭圆图的分析。我们在二维和三维空间中提供了数值实验，以证实我们的理论分析。