Optimal $L^2$ error estimates of unconditionally stable FE schemes for the Cahn-Hilliard-Navier-Stokes system

主讲人：王冀鲁

摘要：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.

主讲人简介：王冀鲁，哈尔滨工业大学（深圳）教授，此前为北京计算科学研究中心特聘研究员。王冀鲁博士的研究兴趣为偏微分方程数值解，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。曾入选国家高层次青年人才计划，目前分别主持和参与国家自然科学基金面上项目和重点项目。

邀请人：李东方

时间：10月13日8:00-10：00