一类奇异与分数阶方程的新型谱方法

Schrodinger equations with inverse-power potentials, second order elliptic equations on a concave domain and Riesz fractional Laplacian equations are all typically singular problems with a profound background in mathematical physics, and have become the frontier research topics of common interests for mathematicians, physicists and chemists. Traditional finite elements, finite differences and spectral methods have only a very low convergence rate for such singular problems, even if various adaptive techniques are adopted. Moreover, finite elements, finite volumes and finite difference are locally supported methods, thus are essentially not competent for seeking numerical solutions of the nonlocal fractional equations. However, evidences support that if the basis functions, schemes, and algorithms can be designed in a spectral method according to the character of singularities, an exponential order of convergence can be still retained. In this propsal, we aim at solving the difficult problems of 3D Schrodinger equations with inverse-power potentials, second-order elliptic equations on a 3D singular domain and the Riesz fractional Laplacian equations in high dimensions. We shall develop some novel spectral and spectral elememt methods of a high accuracy and efficiency, and eventually provide computational mathematics foundations, efficient algorithms and parallel programs for the numerical solution of such an applied problem with corner, edge or surface singularity.

反幂势函数薛定谔方程、奇性区域二阶椭圆方程与Riesz分数阶拉普拉斯方程等奇异问题有着丰富的数学物理背景与内涵，成为数学、物理、化学等领域共同关注的前沿研究课题。对包含凹区域Poisson方程在内的反幂势函数薛定谔方程，无论是传统有限元、有限差分还是经典谱方法，仅能得到非常低的收敛速度，即便采用了各种自适应技术。对分数阶拉普拉斯微分方程而言，有限元、有限体积和有限差分等空间偏导数离散方法本质上均属于局部支持型算法，无法胜任非局部奇性问题的计算。而大量事实证明，若能根据方程与解的奇性特征设计出奇性适应的基函数、格式与算法，则谱方法依然具有指数阶收敛速度。本项目针对三维反幂函数薛定谔方程、三维奇性区域二阶椭圆方程以及高维分数阶拉普拉斯方程这三个极具代表性的难点问题，开展高精度、高效率的新型谱与谱元方法研究，为具有点、线或面奇性的典型应用方程的求解提供计算数学基础、高效共性算法与并行程序原型。

反幂势薛定谔方程、点奇性算子方程与分数阶拉普拉斯方程等奇异问题有着丰富的数学物理背景与内涵，成为数学、物理、化学等领域共同关注的前沿研究课题。无论是采用传统有限元、有限差分还是经典谱方法对这类方程进行离散求解，均仅能得到非常低的有限阶收敛速度，纵使采用各种自适应技术。.本项目围绕这类难题展开，取得了以下研究成果: .(1). 针对分数阶Laplace算子构造了广义Jacobi函数与广义球函数，使得一维区间与任意维球体上对应的分数阶的Poisson方程离散后的刚度矩阵为对角阵，在右端项光滑的情况下谱方法数值解达到指数阶收敛；.(2). 构造了各向同性广义Hermite函数及其伴随Hermite函数，以伴随Hermite函为基函数，构造了任意维全空间分数阶Laplace算子定义的Poisson方程离散后的刚度矩阵为对角阵，在右端项光滑且在无穷远处指数衰减的情况下谱方法数值解达到指数阶收敛。.(5). 构造了Muntz型球函数，设计了有界区域上各类分数阶反幂势薛定谔方程的谱与谱元方法求解格式，数值解达到的最佳指数收敛阶，实现了问题的高效求解；改方法用于Kohn-Sham方程的计算，取得了良好的计算效率。.(4). 构造了Muntz型Hermite函数，用于全空间上各类分数阶反幂势薛定谔方程的谱方法求解，数值离散后的刚度矩阵与质量矩阵为对角或带状矩阵，并真正实现了数值解的最佳指数收敛阶。.(5). 定义了分数次Jacobi与Genbauer多项式，可看成传统Jacobi与Genbauer多项式的插值，选择适当的Sobolev空间，对各类经典奇性函数的谱展开进行分析，首次严格证明了他们的最佳（代数）收敛阶。