報告題目：Adaptive FEM for Helmholtz Equation with Large Wave Number
報 告 人：武海軍 教授
所在單位：南京大學
報告時間：2022年8月13日 星期六 9:00
報告地點：騰訊會議 ID：568-100-135
校内聯系人：呂俊良 lvjl@jlu.edu.cn
報告摘要：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.
報告人簡介：國家傑出青年基金獲得者，南京大學數學系教授、博導。研究領域為偏微分方程數值解法。獲評了江蘇省數學傑出成就獎和南京大學趙世良講座教授，任江蘇省數學會秘書長。
