報告題目：An ultraweak-local discontinuous Galerkin method for PDEs with high order spatial derivatives
報 告 人：徐岩 教授 中國科學技術大學
報告時間：2020年7月13日 10:00-11:00
報告地點：騰訊會議 ID：197 937 207
會議鍊接：https://meeting.tencent.com/s/yajaaeliqrhF
校内聯系人：陶詹晶 zjtao@jlu.edu.cn
報告摘要：
In this paper, we develop a new discontinuous Galerkin method for solving several types of partial differential equations (PDEs) with high order spatial derivatives. We combine the advantages of local discontinuous Galerkin (LDG) method and ultra-weak discontinuous Galerkin (UWDG) method. Firstly, we rewrite the PDEs with high order spatial derivatives into a lower order system, then apply the UWDG method to the system. We first consider the fourth order and fifth order nonlinear PDEs in one space dimension, and then extend our method to general high order problems and two space dimensions. The main advantage of our method over the LDG method is that we have introduced fewer auxiliary variables, thereby reducing memory and computational costs. The main advantage of our method over the UWDG method is that no internal penalty terms are necessary in order to ensure stability for both even and odd order PDEs. We prove stability of our method in the general nonlinear case and provide optimal error estimates for linear PDEs for the solution itself as well as for the auxiliary variables approximating its derivatives. A key ingredient in the proof of the error estimates is the construction of the relationship between the derivative and the element interface jump of the numerical solution and the auxiliary variable solution of the solution derivative. With this relationship, we can obtain the optimal error estimates. The theoretical findings are confirmed by numerical experiments.
報告人簡介：
徐岩，中國科學技術大學數學科學學院教授。2005年于中國科學技術大學數學系獲計算數學博士學位。2005-2007年在荷蘭Twente大學從事博士後研究工作。2009年獲得德國洪堡基金會的支持在德國Freiburg大學訪問工作一年。主要研究領域為高精度數值計算方法。研究工作主要涉及高精度離散格式的設計、分析、及其應用等方面，特别側重于間斷有限元方法及其在流體力學、相場模型、相變問題、水波問題的算法設計、理論分析和應用。2008年度獲全國優秀博士學位論文獎，2017年獲國家自然科學基金委“優秀青年基金”。徐岩教授入選了教育部新世紀優秀人才計劃，主持國家自然科學基金面上項目、德國洪堡基金會研究組合作計劃(Research Group Linkage Programme)、霍英東青年教師基礎研究課題等科研項目。
