华中科技大学高华东博士学术报告

报告题目：Optimal error estimates and recovering technique of a mixed finite element method for nonlinear thermistor equations

报告人：高华东 博士

主持人：安荣  教授

报告时间： 2020年11月23日下午14：30——15：30

腾讯会议：ID 898-267-303

摘 要：This talk is concerned with optimal error estimates and recovering 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 $(\phi, \bm{\theta})$ 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. 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 recovering technique to obtain a new numerical electric potential/field of second-order accuracy. The analysis relies on an $H^{-1}$ norm estimate of the mixed FEM and analysis on a non-classical elliptic map. We provide numerical experiments in both two and three dimensional spaces to confirm our theoretical analyses.

高华东简介

高华东博士，华中科技大学数学与统计学院副教授。分别在香港城市大学数学系(2014年)，南开大学数学科学学院(2011年)和大连理工大学应用数学系(2008年)获得博士，硕士，学士学位。研究方向包括数值分析: 微分方程数值解, 有限元方法与差分方法, 尤其是对非线性抛物问题的数值求解与分析; 数学建模与计算物理: 多孔介质中热和水汽的传导流动, 计算超导现象, 计算微磁学, 计算电热学。目前主持面上基金一项，已正式发表论文约二十篇。