报 告 人：李会元
报告题目：Novel Spectral/Spectral Element Methods for Some Singular Eigenvalue Problems
报告时间：2017年11月28日（周二）下午16:30
报告地点：静远楼204报告厅
主办单位：数学与统计学院、科技处
报告人简介：
李会元，中国科学院软件研究所研究员，博士生导师。主要研究领域为大型异构系统下的高性能科学计算与数学软件，数值偏微分方程的谱方法与谱元素法，特征值问题的高性能计算方法等. 在国际著名期刊SIAM J. Numer. Anal., Math. Comp., J.of Sci.Comp.等上发表论文四十余篇，主持多项国家自然科学基金面上项目.
报告摘要：
We propose and analyze some efficient spectral/spectral element methods to solve singular eigenvalue problems related to the Schrodinger operator with an inverse-power potential.
For the Schrodinger eigenvalue problem $-\Delta u +V(x)u=\lambda u$ with a critical (inverse-square) potential $V(x)=c_2|x|^{-2}$, there are three stages in our investigation: We start from a ball of any dimension, and design two novel spectral methods by modifying the polynomial basis to fit the singularities of the eigenfunctions. At the second stage, we move to circular sectors in the two-dimensional setting. At the final stage, we extend the idea to arbitrary polygonal domains. We propose a mortar spectral element approach: a polygonal domain is decomposed into several subdomains with each singular corner including the origin covered by a circular sector, in which origin and corner singularities are handled similarly as in the former stages, and the remaining domains are either a standard quadrilateral/triangle or a quadrilateral/triangle with a circular edge, in which the traditional polynomial based spectral method is applied. All subdomains are linked by mortar elements.
Next, an efficient spectral method for the Schrodinger eigenvalue problem with a regular potential $V(x)=c_1|x|^{-1}$ on a ball of any dimension by adopting the Sobloev-orthogonal basis functions with respect to the Laplacian operator to overwhelm the homogeneous inverse potential and to eliminate the singularity of the eigenfunctions. Then weextend this spectralmethod to arbitrarypolygonal domains again by the mortar element method with each corner covered by a circular sector and origin covered by a circular disc.
Furthermore, for the Schrodinger eigenvalue problem with a singular potential $V(x)=c_3|x|^{-3}$, we devise a novel spectral method by modifyingtheformer Sobloev-orthogonal bases to fitthe stronger singularity. As in the case of $|x|^{-1}$ potential, this approach can be extended to arbitrary polygonal domains by the Mortar element method as well.
Finally, forthe singularelliptic eigenvalue problem $-\frac{\partial^2}{\partialx^2}u-\frac{1}{x^2}\frac{\partial^2}{\partial y^2}u =\lambda u$ on rectangles, we propose a novel spectral method by using tensorial bases composed of the $L^2$- and $H^1$-simultaneously orthogonal functions in the $y$-direction and the Sobolev-orthogonal functions with respect to the Schrodinger operator with an inverse-square potential in the $x$-direction.
Numericalexperimentsindicate that all our methods possess exponential orders of convergence, and are superior to the existing polynomial based spectral/spectral element methods and hp-adaptive methods.
