交界面椭圆、抛物和Maxwell方程的有限元分析和进展 献给林群教授80华诞

Advances of finite element analysis for elliptic, parabolic and Maxwell interface equations

本文综述交界面问题的有限元离散数值方法近十几年来的最新进展,重点讨论椭圆型、抛物型和Maxwell型交界面问题的有限元方法,总结已有的相关研究结果,如正则性、变分形式、有限元离散策略、数值积分策略、交界面逼近及其对离散解的影响和离散解的收敛性分析等.伴随历史发展,本文还梳理了交界面问题的发展脉络.最后列举一个交界面模型的实际应用,即不可测热交换界面上热通量重构问题,着重分析自适应有限元方法的后验误差估计.

The development of finite element methods for interface problems in the recent two decades is re-viewed with an emphasis on elliptic, parabolic and Maxwell interface problems. We summarize some important results, e.g., regularity, variational forms, discretization schemes, numerical quadrature, interface approxima- tion, and convergence analysis. The state of the art for these problems is described along with the history of developments. Finally, we outline some applications governed by the PDE systems with interfaces.