报告人：易利军（上海师范大学）

邀请人：王海永

报告时间：2022年 3月31日 （星期四）10: 00-11:30  

报告地点：腾讯会议：515 339 138  会议密码：0331

报告题目：Superconvergence and postprocessing of the continuous Galerkin method for nonlinear initial value problems with smooth and singular solutions

报告摘要：In this talk, we introduce a very simple but efficient postprocessing technique for improving the accuracy of the continuous Galerkin (CG) time stepping method for nonlinear first-order initial value problems with smooth and singular solutions. The key idea of the postprocessing technique is to add a higher order polynomial of degree k+1 to the k-th CG solution. We first prove that the CG method superconverges at the nodal points of an arbitrary time partition. We further establish global superconvergence error bounds for the postprocessed CG approximations over arbitrary time partitions, and particularly, we show that the convergence rates of the L^2-, H^1- and L^\infty-estimates for regular solutions over quasi-uniform meshes are improved by one order. As a by-product of the postprocessed superconvergence results, we construct asymptotically exact a posteriori error estimators and prove that they converge to the true errors under mesh refinement. Moreover, for solutions with initial singularities, we prove that optimal algebraic convergence rates can be obtained for the CG method on graded meshes. We further show that, after postprocessing, the convergence rates of the L^2-, H^1- and L^\infty-estimates for singular solutions over graded meshes are also improved by one order. Numerical examples are presented to illustrate the theoretical results.

报告人简介：易利军，上海师范大学数学系教授，博士生导师，研究方向为积分和微分方程数值解。在计算数学和应用数学领域的顶级期刊《SIAM J. Numer. Anal.》、《Math. Comp.》和《Math. Models Methods Appl. Sci.》等发表SCI论文30余篇。