报告一：Efficient and accurate structure preserving schemes for complex nonlinear systems

报告时间：2024年9月26日（星期四）9:00-10:00

报告地点：翡翠科教楼B座1710

报告人：沈捷 教授

工作单位：宁波东方理工大学

主办单位：数学学院

报告简介：Many complex nonlinear systems have intrinsic structures such as energy dissipation or conservation, and/or positivity/maximum principle preserving. It is desirable, sometimes necessary, to preserve these structures in a numerical scheme.

I will present some recent advances on using the scalar auxiliary variable (SAV) approach and Lagrange multiplier approach to develop highly efficient and accurate structure preserving schemes for a large class of complex nonlinear systems. These schemes can preserve energy dissipation/conservation as well as other global constraints and/or are positivity/bound preserving, only require solving decoupled linear equations with constant coefficients at each time step, and can achieve higher-order accuracy.

However, a direct application of the SAV approach will not preserve positivity or maximum principle. In the second lecture,I will then present two strategies, one based on convex splitting and the other based on SAV with transformation,to construct efficient energy stable and positivity preserving schemes for certain nonlinear evolution systems, such as the Poisson-Nernst-Planck (PNP) equation and Keller-Segel equation, whose solutions remain to be positive.

报告人简介：沈捷教授现为宁波东方理工大学讲席教授和数学科学学院院长。他于1982年毕业于北京大学计算数学专业, 1987年获法国巴黎十一大学博士学位，2009年入选海外高层次人才项目，2017年当选 AMS fellow，2020当选 SIAM fellow，2023年全职回国前为普渡大学杰出教授。

沈捷教授主要从事偏微分方程数值解研究工作，具体研究方向包括谱方法数值分析理论, 计算流体, 以及计算材料科学。在国际杂志上发表了两百多篇论文，两本专著，在谷歌学术的引用逾2万八千余次。

报告二：C1-conforming Petrov-Galerkin methods for 2nd-order elliptic problems

报告时间：2024年9月26日（星期四）10:00-11:00

报告地点：翡翠科教楼B座1710

报告人：张智民 教授

工作单位：韦恩州立大学

主办单位：数学学院

报告简介：In certain applications,2nd-order elliptic problems lack divergent forms due to regularity restrictions, necessitating the direct discretization of the2nd-order derivatives. In this work, we develop a new Petrov-Galerkin method that employs a C1-conforming finite element for the trial space and an L2-discontinuous element for the test space. We demonstrate that the numerical solution obtained through this new method converges to the exact solution with an order of 2k-2(where k > 2 is the polynomial degree) at the nodal points for both function value and the gradient, assuming a rectangular mesh.

报告人简介：张智民，中国科学技术大学学士（1982）硕士（1985）、马里兰大学（University of Maryland，College Park）博士（1991）、韦恩州立大学(Wayne State University)教授（2002-）、教育部“长江学者”（2010）、国家引进海外高层次人才（2012），现任和曾任10个国内外数学杂志编委，包括Mathematics of Computation（2009-2017），Journal of Scientific Computing（2011-2017），Numerical methods for Partial Differential Equations（2013-），Communications on Applied Mathematics and Computation（2019-），CSIAM Transaction on Applied Mathematics（2019-），《数学文化》（2010-）等,发表SCI论文200余篇。

张智民教授长期从事计算方法，所提出的多项式保持重构（Polynomial Preserving Recovery—PPR）方法2008年被大型商业软件COMSOL Multiphysics采用并沿用至今。计算物理等杂志的编委。