报告人：邱蔚峰 教授（香港城市大学）

报告人简介：邱蔚峰，香港城市大学教授，于2000年获得上海师范大学学士学位，2006年获得阿拉巴马大学亨茨维尔分校硕士学位，2010年获得德克萨斯大学奥斯汀分校博士学位。他的博士生导师是Leszek Demkowicz教授。他的博士后导师是Bernardo Cockburn教授。在2012年加入城市大学之前，他曾在明尼苏达大学IMA（数学及其应用研究所）担任博士后研究员。他的主要研究兴趣包括PDE的科学计算和数值分析。

（一）报告题目：Analysis of a mixed finite element method for the quad-curl problem

报告摘要：Quad-curl term is essential in the resistive magnetohydrodynamics (MHD) equation and the  fourth order inverse electromagnetic scattering problem. It is desirable to develop simple and efficient numerical methods for the quad-curl problem. In this paper, we firstly generalize some regularity results for the quad-curl problem on Lipschitz polyhedron domains. Then, we propose a mixed finite element method for solving the quad-curl problem. With a novel discrete Sobolev imbedding inequality for the piecewise polynomials, we obtain stability results and derive optimal error estimates relying on a low regularity assumption of the exact solution.

报告时间：2019年6月11日（星期二）晚上19:00

报告地点：科技楼南楼702室

（二）报告题目：Parameter-free superconvergent H(div)-conforming HDG methods for the Brinkman equations

报告摘要：In this paper, we present new parameter-free {superconvergent} H(div)-conforming HDG methods for the Brinkman equations on both simplicial and rectangular meshes. The methods are based on a velocity gradient-velocity-pressure formulation, which can be considered as a natural extension of the H(div)-conforming HDG method (defined on simplicial meshes for the Stokes flow [Math. Comp. 83(2014), pp. 1571-1598].We obtain optimal  L2-error estimate for the velocity in both the Stokes-dominated regime (high viscosity/permeability ratio) and Darcy-dominated regime (low viscosity/permeability ratio). We also obtain superconvergent L2-estimate of one order higher for a suitable projection of the velocity error in the Stokes-dominated regime. Moreover, thanks to H(div)-conformity of the velocity, our velocity error estimates are independent of the pressure regularity. Furthermore, we  provide a discrete H1-stability result of the velocity field, which is essential in the error analysis of the natural generalization of these new HDG methods to the incompressible Navier-Stokes equations.Preliminary numerical results on both triangular and rectangular meshes in two dimensions confirm our theoretical predictions.

报告时间： 2019年6月12日（星期三）上午10:30

报告地点：科技楼南楼702室