Adaptive First-Order System Least-Squares Finite Element Methods for Second Order Elliptic Equations in Non-Divergence Form

：邱蔚峰（香港城市大学）
：2022-03-10 15:00
：腾讯会议ID：267133365（无密码）

报告人：邱蔚峰（香港城市大学）

时  间：3月10日下午15:00

地  点：腾讯会议ID：267133365（无密码）

内容摘要:

In this talk we will introduce adaptive first-order system least-squares finite element methods for second-order elliptic partial differential equations in non-divergence form. Unlike the classical finite element method which uses weak formulations of PDEs not applicable for the non-divergence equation, the first-order least-squares formulations naturally have stable weak forms without using integration by parts, allow simple finite element approximation spaces, and have build-in a posteriori error estimators for adaptive mesh refinements. The non-divergence equation is first written as a system of first-order equations by introducing the gradient as a new variable. Then two versions of least-squares finite element methods using simple C0 finite elements are developed in the paper, one is the L2-LSFEM which uses linear elements, the other is the weighted-LSFEM with a mesh-dependent weight to ensure the optimal convergence. Under a very mild assumption that the PDE has a unique solution, optimal a priori and a posteriori error estimates are proved. With an extra assumption on the operator regularity which is weaker than traditionally assumed, convergences in standard norms for the weighted-LSFEM are also discussed. $L^2$-error estimates are derived for both formulations. We perform extensive numerical experiments for smooth, non-smooth, and even degenerate coefficients on smooth and singular solutions to test the accuracy and efficiency of the proposed methods.

个人简介：

Prof. Weifeng Qiu received his BSc from Shanghai Normal University in 2000, Master from University of Alabama in Huntsville in 2006, and PhD from the University of Texas at Austin in 2010. His PhD advisor is Professor Leszek Demkowicz. Before joining City University in 2012, he worked as a Postdoctoral Fellow at IMA (Institute for Mathematics and Its Applications), University of Minnesota. His postdoctoral mentor is Professor Bernardo Cockburn. His major research interests include scientific computing and numerical analysis for PDE.

 

联系人：陈黄鑫