题目一：Optimal error estimates of linearized FEMs for fractional problems

内容简介：Several linearized L1-Galerkin finite element methods are proposed to solve the multi-dimensional nonlinear time fractional parabolic and Schrodinger equations. In terms of a temporal-spatial error splitting argument, we prove that the finite element approximations in certain norms are bounded without any time stepsize conditions. More importantly, by using the recent discrete fractional Gronwall type inequality, optimal error estimates of the numerical schemes are obtained unconditionally, while the classical analysis for multi-dimensional nonlinear fractional problems always required certain time-step restrictions dependent on the spatial mesh size. Numerical examples are given to illustrate our theoretical results.

报告人：华中科技大学大学李东方教授

报告人简介：华中科技大学数学与统计学院教授，中国系统仿真学会仿真算法专业委员会委员。主要从事微分方程数值解、系统仿真和信号处理等方面的研究。曾先后赴加拿大McGill大学，香港城市大学从事博士后研究。截至目前在《SIAM. J. Numer. Anal.》，《SIAM. J. Sci. Comput.》、《J. Comp. Phys.》、《Appl. Comp. Harm. Appl.》等多个国际著名计算学科SCI期刊上发表论文40余篇。主持国家自然科学基金面上项目、青年基金各一项，博士后基金一项，参与多项国家自然科学基金。先后获得华中科技大学学术新人奖、香江学者奖等。

题目二：Efficient and accurate Fourier approximations of nonlocal diffusion models

内容简介：This work is concerned with the Fourier spectral methods for the numerical approximation of integral differential equation models associated with some linear nonlocal diffusion operators with periodic boundary conditions. For radially symmetric kernels, the nonlocal operators under consideration are diagonalizable in Fourier space. A challenge is the accurate and fast evaluation of the Fourier symbols which consist of possibly singular and highly oscillatory integrals. For a large class of fractional power-like kernels, we propose a new approach to reformulate the Fourier symbols as coefficients of a series expansion which correspond to solutions of some simple ODE models. We then propose a hybrid algorithm to provide fast evaluation of Fourier symbols in both one and higher dimensional spaces. On the other hand, we also propose a splitting technique to handle the non-smooth solutions of linear nonlocal diffusion models. As applications, we combine this hybrid spectral discretization in the spatial variables and fourth-order exponential time differencing Runge--Kutta for temporal discretization to approximate some nonlocal gradient flows including nonlocal Allen--Cahn equation, nonlocal Cahn-Hilliard equation, and nonlocal phase-field crystal models. Numerical results show the accuracy and effectiveness of this fully discrete schemes and illustrate some interesting phenomena associated with the nonlocal models.

报告人：南方科技大学杨将副教授

报告人简介：2010年本科毕业于浙江大学，2014年于香港浸会大学数学系获得博士学位。之后分别于2014/08-2015/08和2015/08-2017/07在美国宾夕法尼亚州立大学和美国哥伦比亚大学从事博士后研究工作。2017/07开始在南方科技大学数学系工作，现为副教授。入选第十四批“千人计划”青年项目。主要研究的领域偏微分方程数值解，特别是关于相场模型以及非局部模型的计算与研究。在计算数学相关的国际期刊发表了数十篇论文。

时间：2018年10月19日（周五）下午2：00始

地点：南海楼338室

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2018年10月17日