荔园数学讲坛: An approximation theory for nonlinear problems and its application to the Schroedinger-Poisson model

主讲人：中国科学院数学与系统科学研究院 郑伟英 研究员

时间：2023年4月11日9:00–10:00

地点：腾讯会议310-599-630 密码：1234

链接：https://meeting.tencent.com/dm/KlZlo4sa9T7A 

内容摘要： 

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: 

郑伟英，中国科学院数学与系统科学研究院研究员。1996年本科毕业于郑州大学，2002年博士毕业于北京大学，2017年获国家杰出青年科学基金资助，2019-2021年任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，现任中国数学会计算数学分会常务理事、中国工业与应用数学学会副秘书长等。主要从事电磁场计算方法及理论、半导体器件的计算方法和理论的研究。目前担任J. Comput. Math., Adv. Appl. Math. Mech.等杂志的编委。