内容簡介：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.

報告人：王冀魯

報告人簡介：哈爾濱工業大學（深圳）教授，博導，曾入選國家級青年人才計劃，此前為北京計算科學研究中心特聘研究員。她的研究課題主要集中在偏微分方程數值解，具體包括關于淺水波方程、多孔介質中不可壓混溶驅動模型、薛定谔方程以及分數階方程的數值方法，研究成果發表在 《Numer. Math》、《SIAM J. Numer. Anal.》、《Math. Comput.》、《SIAM J. Control Optim.》等計算數學權威期刊，目前主持國家自然科學基金面上項目、深圳市傑出青年研究項目等。

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

内容簡介：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.

報告人：李東方

報告人簡介：華中科技大學數學與統計學院教授，博導，國家級青年人才。主持國家級課題5項。主要從事微分方程數值解、機器學習和信号處理等領域的研究工作。尤其在微分方程保結構算法和分數階微分方程的高效數值算法和理論上取得一些有意義的進展。相關工作發表在《SIAM. J. Numer. Anal.》，《SIAM. J. Sci. Comput.》、《Math. Comput》、《J. Comp. Phys.》等多個國際著名計算學科SCI期刊上，多篇為高被引論文。

時  間：2024年9月19日（周四）下午15：00開始

地  點：騰訊會議：499-409-163

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