报告人：王冀鲁 研究员（北京计算科学研究中心）

报告人简介：王冀鲁，北京计算科学研究中心研究员，主要研究方向为PDE的科学计算和数值分析，尤其是有限元方法的理论分析。她的学术工作发表在《SIAM J. Numer. Anal.》《Numer. Math.》《Math. Comput.》等著名计算数学SCI杂志上。

（一）报告题目：Convergence analysis of Crank-Nicolson Galerkin FEMs for a nonlinear Schr\"{o}dinger-Helmholtz system

报告摘要：In this work, we consider the unconditional stability and optimal $L^2$ error estimates of linearized Crank-Nicolson Galerkin FEMs for a nonlinear Schr\"{o}dinger-Helmholtz system in $\mathbb{R}^d$ ($d=2,3$). By introducing a corresponding time-discrete system, we separate the error into two parts, i.e., the temporal error and the spatial error. Since the latter is $\tau$-independent, the uniform boundedness of numerical solutions in $L^{\infty}$-norm follows an inverse inequality immediately without any restrictions on time stepsize. Then, optimal error estimates are obtained in a routine way. Numerical examples in both two and three dimensional spaces are given to illustrate our theoretical results.

报告时间：2019年6月12日（星期三）晚上19:20

报告地点：科技楼南楼702室

（二）报告题目：Unconditionally convergent $L1$-Galerkin FEMs for nonlinear time-fractional Schr\"{o}dinger equations

报告摘要：In this work, a linearized $L1$-Galerkin finite element method is proposed to solve the multi-dimensional nonlinear time-fractional Schr\"{o}dinger equation. In terms of a temporal-spatial error splitting argument,we prove that the finite element approximations in $L^2$-norm and $L^\infty$-norm are bounded without any time stepsize conditions. More importantly, by using a 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.

报告时间：2019年6月13日（星期四）上午10:20

报告地点：科技楼南楼602室