## 数学与统计学院学术报告 — 李步扬教授

应数学与统计学院邓伟华教授的邀请，香港理工大学应用数学系助理教授李步扬博士将于近期访问我校，期间将做学术报告。
题  目 (一)：Runge-Kutta time discretization of nonlinear parabolic equations studied via discrete maximal parabolic regularity
时         间：2018年1月15日9:00
地         点：齐云楼911室
报 告 摘 要：For a large class of fully nonlinear parabolic equations, which include gradient flows for energy functionals that depend on the solution gradient, the semidiscretization in time by implicit Runge–Kutta methods such as the Radau IIA methods of arbitrary order is studied. Error bounds are obtained in the $W^{1,\infty}$ norm uniformly on bounded time intervals and, with an improved approximation order, in the parabolic
energy norm. The proofs rely on discrete maximal parabolic regularity.
This is used to obtain $W^{1,\infty}$ estimates, which are the key to the numerical analysis of these problems.

题  目 (二)：Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint
时         间：2018年1月16日9:00
地         点：齐云楼911室
报 告 摘 要：In this work, we present numerical analysis for a distributed optimal control problem, with box constraint on the control, governed by a subdiffusion equation which involves a fractional derivative of order $\alpha\in(0,1)$ in time. The fully discrete scheme is obtained by applying the conforming linear Galerkin finite element method in space, with L1 scheme or backward Euler convolution quadrature in time, and the control variable by a variational type discretization. With a space mesh size $h$ and time stepsize $\tau$, we prove the following order of convergence for the numerical solutions of the optimal control problem: $O(\tau^{\min(1/2+\alpha-\varepsilon,1)}+h^2)$ in the discrete  $L^2(0,T;L^2(\Omega))$ norm and $O(\tau^{\alpha-\varepsilon}+l_{h}^2h^2)$ in the discrete $L^{\infty}(0,T;L^2(\Omega))$ norm, with an arbitrarily small positive number $\varepsilon$ and a logarithmic factor $l_h=\ln(2+1/h)$. Numerical experiments are provided to support the theoretical results.

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李步扬博士于2005年在山东大学取得数学学士学位，并分别于2007、2009 及2012年在香港城市大学取得应用数学硕士、哲学硕士及博士学位。李博士于2012年12月开始任职于南京大学，并于2015年7月晋升为副教授。在2015年 6月至2016年5月期间，李博士在德国图宾根大学兼任洪堡学者的工作。李博士2016年6月加入香港理工大学应用数学系担任助理教授一职。李博士当前的主要研究方向是偏微分方程的数值解法和数值分析，在SIAM J. Numer. Anal., SIAM J. Sci. Comput., Math. Comput., Numer. Math.  等计算数学顶级期刊上发表论文40多篇。

数学与统计学院
萃英学院
二〇一八年一月十二日