An approximation theory for nonlinear problems and its application to the Schroedinger-Poisson model
数学公众报告(120周年校庆系列第74场)
报告题目(Title)：An approximation theory for nonlinear problems and its application to the Schroedinger-Poisson model
报告人(Speaker)：郑伟英（中国科学院数学与系统科学研究院，计算数学与科学工程计算研究所）
地点(Place)：后主楼1124
时间(Time)：2023年3月3日(周五),10:30-11:30
邀请人(Inviter)：陈华杰
报告摘要
We present a unified theory of error estimate for the Galerkin approximation of a class of nonlinear problems. The three conditions for the error estimate are that, 1) the original problem has a solution u which is the fixed point of a compact operator A, 2) A is Frechet-differentiable at u and I-A'[u] has a bounded inverse in a neighbourhood of u, and 3) the Galerkin approximation of the problem defines an approximate operator A_h which is continuous and converges uniformly to A in the neighbourhood of u. The theory states that the approximate problem has a solution u_h in the neighbourhood of u and gives the estimate of u-u_h in terms of the approximation parameter h. The superiority of the theory is that no assumptions are made about the well-posedness of the approximate problem.
The theory is applied to two kinds of approximations of the nonlinear Schroedinger-Poisson model. The first approximation is made by truncating the series of the electron density into the sum of a finite number of remained terms. We prove that the approximate solution converges exponentially to the exact solution with respect to the truncated eigenvalue. The second approximation is to solve the Schroedinger-Poisson model with the linear Lagrangian finite element method. We prove the optimal error estimate between the numerical solution and the exact solution. As far as we know, the error estimate for the Schroedinger-Poisson model is new in the literature.
主讲人简介
中国科学院数学与系统科学研究院研究员，主要从事电磁场和半导体器件的计算方法和理论等方面的研究。 2017年获国家杰出青年科学基金资助，2019-2021年被聘为中科院数学与系统科学研究院“冯康首席研究员”，2021年获“冯康科学计算奖”。
