高效求解包含反平方势的薛定谔方程本征值问题的谱与谱元方法
Efficient Spectral and Spectral Element Methods for Eigenvalue Problems of Schrödinger Equations with an Inverse Square Potential
AI总结：
本文提出针对包含反平方势的薛定谔方程本征值问题的高效谱与谱元方法，通过改进基函数以适应奇异性和在多边形域中处理尖角问题，实现指数收敛性。
摘要：
本文研究了薛定谔算子$-Δu + \frac{c^2}{|x|^2}u$的本征值问题的数值近似。我们的研究分为三个阶段：首先从任意维数的球体开始，其中径向方向的精确解可通过分数次贝塞尔函数表达。这一知识帮助我们设计了两种新的谱方法，通过修改多项式基函数以适应本征函数的奇点。第二阶段转向二维圆扇域，径向方向同样可用分数次贝塞尔函数表示，仅在切向方向需要从第一阶段进行修改。最后，我们将方法扩展到任意多边形域，提出了一种mortar谱元方法：将多边形域分解为若干子域，每个包含原点的奇异角被覆盖为圆扇域，原点和角的奇点处理方式与前几阶段相同，其余域为标准四边形/三角形或带有圆边的四边形/三角形，应用传统多项式基的谱方法。所有子域通过mortar元素连接（注意我们可能有悬挂节点）。在所有三个阶段中均实现了指数收敛率。数值实验表明，我们的新方法优于传统多项式基的谱（或谱元）方法和$hp$-自适应方法。我们的研究提供了一种新的有效方法，用于处理包含拉普拉斯算子的薛定谔算子在具有内角的多边形域中的本征值问题。
英文摘要：
In this article, we study numerical approximation of eigenvalue problems of the Schrödinger operator $\displaystyle -Δu + \frac{c^2}{|x|^2}u$. There are three stages in our investigation: We start from a ball of any dimension, in which case the exact solution in the radial direction can be expressed by Bessel functions of fractional degrees. This knowledge helps us to 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. Again the radial direction can be expressed by Bessel functions of fractional degrees. Only in the tangential direction some modifications are needed from stage one. 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 sub-domains 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 sub-domains are linked by mortar elements (note that we may have hanging nodes). In all three stages, exponential convergence rates are achieved. Numerical experiments indicate that our new methods are superior to standard polynomial based spectral (or spectral element) methods and $hp$-adaptive methods. Our study offers a new and effective way to handle eigenvalue problems of the Schrödinger operator including the Laplacian operator on polygonal domains with reentrant corners.
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