报告题目：Adaptive FEM for Helmholtz Equation with Large Wave Number

报 告 人：武海军 教授

所在单位：南京大学

报告时间：2022年8月13日 星期六 9:00

报告地点：腾讯会议 ID：568-100-135

校内联系人：吕俊良 lvjl@jlu.edu.cn

报告摘要：A posteriori upper and lower bounds are derived for the linear finite element method (FEM) for the Helmholtz equation with large wave number. It is proved rigorously that the standard residual type error estimator seriously underestimates the true error of the FE solution for the mesh size $h$ in the preasymptotic regime, which is first observed by Babu\v{s}ka,~et~al. for an one dimensional problem. By establishing an equivalence relationship between the error estimators for the FE solution and the corresponding elliptic projection of the exact solution, an adaptive algorithm is proposed and its convergence and quasi-optimality are proved under condition that $k^3h_0^{1+\alpha}$ is sufficiently small, where $k$ is the wave number, $h_0$ is the initial mesh size, and $\frac12<\alpha\le 1$ is a regularity constant depending on the maximum reentrant angle of the domain. Numerical tests are given to verify the theoretical findings and to show that the adaptive continuous interior penalty finite element method (CIP-FEM) with appropriately selected penalty parameters can greatly reduce the pollution error and hence the residual type error estimator for this CIP-FEM is reliable and efficient even in the preasymptotic regime.

报告人简介：国家杰出青年基金获得者，南京大学数学系教授、博导。研究领域为偏微分方程数值解法。获评了江苏省数学杰出成就奖和南京大学赵世良讲座教授，任江苏省数学会秘书长。