Accuracy-enhancement of discontinuous Galerkin methods for PDEs containing high order spatial derivatives
数学公众报告(120周年校庆系列第47场)
报告题目(Title)：Accuracy-enhancement of discontinuous Galerkin methods for PDEs containing high order spatial derivatives
报告人(Speaker)：徐岩 教授（中国科学技术大学）
地点(Place)：腾讯会议ID：388600151
时间(Time)：2022年10月21日（周五） 15:00-16:00
邀请人(Inviter)：曹外香
报告摘要
In this talk, we consider the accuracy-enhancement of discontinuous Galerkin (DG) methods for solving partial differential equations (PDEs) with high order spatial derivatives. It is well known that there are highly oscillatory errors for finite element approximations to PDEs that contain hidden superconvergence points. To exploit this information, a Smoothness-Increasing Accuracy-Conserving (SIAC) filter is used to create a superconvergence filtered solution. This is accomplished by convolving the DG approximation against a B-spline kernel. Previous theoretical results about this technique concentrated on first- and second-order equations. However, for linear higher order equations, Yan and Shu numerically demonstrated that it is possible to improve the accuracy order to 2k+1 for local discontinuous Galerkin (LDG) solutions using the SIAC filter. In this work, we firstly provide theoretical proof for this observation. Furthermore, we prove the accuracy order of the ultra-weak local discontinuous Galerkin (UWLDG) solution could be improved to 2k+2-m using the SIAC filter, where m=n/2, n is the order of PDEs. Finally, we computationally demonstrate that for nonlinear higher order PDEs, we can also obtain a superconvergence approximation with the accuracy order of 2k+1 or 2k+2-m by convolving the LDG solution and the UWLDG solution against the SIAC filter, respectively.
主讲人简介
徐岩，中国科学技术大学数学科学学院教授。2005年于中国科学技术大学数学系获计算数学博士学位。2005-2007年在荷兰Twente大学从事博士后研究工作。2009年获得德国洪堡基金会的支持在德国Freiburg大学访问工作一年。主要研究领域为高精度数值计算方法。2008年度获全国优秀博士学位论文奖，2017年获国家自然科学基金委“优秀青年基金”。徐岩教授入选了教育部新世纪优秀人才计划，主持国家自然科学基金面上项目、德国洪堡基金会研究组合作计划(Research Group Linkage Programme)、霍英东青年教师基础研究课题等科研项目。担任Journal of Scientific Computing, Advances in Applied Mathematics and Mechanics, Communication on Applied Mathematics and Computation、计算物理等杂志的编委。
