内容简介：The paper is concerned with the analysis of a popular convex-splitting finite element method for the Cahn-Hilliard-Navier-Stokes system, which has been widely used in practice. Since the method is based on a combined approximation to multiple variables involved in the system, the approximation to one of the variables may seriously affect the accuracy for others. Optimal-order error analysis for such combined approximations is challenging. The previous works failed to present optimal error analysis in $L^2$-norm due to the weakness of the traditional approach. Here we first present an optimal error estimate in $L^2$-norm for the convex-splitting FEMs. We also show that optimal error estimates in the traditional (interpolation) sense may not always hold for all components in the coupled system due to the nature of the pollution/influence from lower-order approximations. Our analysis is based on two newly introduced elliptic quasi-projections and the superconvergence of negative norm estimates for the corresponding projection errors. Numerical examples are also presented to illustrate our theoretical results. More important is that our approach can be extended to many other FEMs and other strongly coupled phase field models to obtain optimal error estimates.

报告人：王冀鲁

报告人简介：哈尔滨工业大学（深圳）教授，博导，曾入选国家级青年人才计划，此前为北京计算科学研究中心特聘研究员。她的研究课题主要集中在偏微分方程数值解，具体包括关于浅水波方程、多孔介质中不可压混溶驱动模型、薛定谔方程以及分数阶方程的数值方法，研究成果发表在 《Numer. Math》、《SIAM J. Numer. Anal.》、《Math. Comput.》、《SIAM J. Control Optim.》等计算数学权威期刊，目前主持国家自然科学基金面上项目、深圳市杰出青年研究项目等。

题目二：Energy-Decaying  Methods for the phase-field models

内容简介：We construct and analyze a class of linearized Runge--Kutta (RK) methods, which can be of arbitrarily high order, for the time discretization of the phase field equations. We prove that the proposed s-stage  methods have  sth-order convergence in time and satisfy a discrete version of the energy decay property. Numerical examples are provided to illustrate the discrete energy decay property and accuracy of the proposed methods.

报告人：李东方

报告人简介：华中科技大学数学与统计学院教授，博导，国家级青年人才。主持国家级课题5项。主要从事微分方程数值解、机器学习和信号处理等领域的研究工作。尤其在微分方程保结构算法和分数阶微分方程的高效数值算法和理论上取得一些有意义的进展。相关工作发表在《SIAM. J. Numer. Anal.》，《SIAM. J. Sci. Comput.》、《Math. Comput》、《J. Comp. Phys.》等多个国际著名计算学科SCI期刊上，多篇为高被引论文。

时  间：2024年9月19日（周四）下午15：00开始

地  点：腾讯会议：499-409-163

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信息科学技术学院/网络空间安全学院

2024年9月18日