Efficient Spectral Sparse Grid Methods and Applications to High-Dimensional Elliptic Equations II. Unbounded Domains

This is the second part in a series of papers on using spectral sparse grid methods for solving higher-dimensional PDEs. We extend the basic idea in the first part [18] for solving PDEs in bounded higher-dimensional domains to unbounded higher-dimensional domains, and apply the new method to solve the electronic Schrödinger equation. By using modified mapped Chebyshev functions as basis functions, we construct mapped Chebyshev sparse grid (MCSG) methods which enjoy the following properties: (i) the mapped Chebyshev approach enables us to build sparse grids with Smolyak’s algorithms based on nested, spectrally accurate quadratures, and allows us to build fast transforms between the values at the sparse grid points and the corresponding expansion coefficients; (ii) the mapped Chebyshev basis functions lead to identity mass matrices and very sparse stiffness matrices for problems with constant coefficients, and allow us to construct a matrix-vector product algorithm with quasi-optimal computational cost even for problems with variable coefficients; and (iii) the resultant linear systems for elliptic equations with constant or variable coefficients can be solved efficiently by using a suitable iterative scheme. Ample numerical results are presented to demonstrate the efficiency and accuracy of the proposed algorithms.