A novel spectral approximation and error estimation for transmission eigenvalues in spherical domains
数学公众报告(120周年校庆系列第53场)
报告题目(Title)：A novel spectral approximation and error estimation for transmission eigenvalues in spherical domains
报告人(Speaker)：安静 教授（贵州师范大学）
地点(Place)：腾讯会议ID：853322203
时间(Time)：2022年11月4日（周五）15:00-16:00
邀请人(Inviter)：曹外香
报告摘要
In this paper, we propose and analyze an efficient spectral-Galerkin method based on a mixed formulation with dimension reduction for the Helmholtz transmission eigenvalue problem in spherical domains. By introducing an auxiliary function, we rewrite the original problem as an equivalent fourth-order coupled form in spherical coordinates. Using properties of the spherical harmonic and Laplace-Beltrami operators, we further decompose the original problem into a series of one-dimensional fourth-order coupled linear eigenvalue problems, for which a new mixed variational formulation and its discretization is developed. For error estimates of numerical eigenvalues and eigenfunctions, we recall the spectral theory of compact operators. Towards this end, we derive the essential polar conditions, define a class of weighted Sobolev spaces, and most importantly, prove a sequence of two compact embedding properties for the weighted Sobolev spaces, based on which the spectral theory of compact operators for the variational formulation and discrete system can be established. Finally, some numerical examples are presented to confirm the theoretical error analysis and the efficiency of our algorithm.
主讲人简介
安静，贵州师范大学数学科学学院教授，中国数学会计算数学分会第十届委员。主要从事偏微分方程及其特征值问题的理论和数值计算方面的研究，研究问题包括传输特征值问题，非线性哈密顿系统，电磁场方程等。曾先后赴美国普渡大学、新加坡南洋理工大学、中国科学软件研究所做访问学者。2016年1月-2018年12月先后于北京计算科学研究中心从事博士后研究和天元数学访问学者。截至目前主持完成国家自然科学基金项目2项，在研国家自然科学基金项目1项，主持完成贵州省科技厅基金项目2项，2018年获得贵州省人民政府特殊津贴，2021获得贵州省自然科学三等奖。
