Efficient Spectral Sparse Grid Methods and Applications to High-Dimensional Elliptic Problems

We develop in this paper some efficient algorithms which are essential to implementations of spectral methods on the sparse grid by Smolyak’s construction based on a nested quadrature. More precisely, we develop a fast algorithm for the discrete transform between the values at the sparse grid and the coefficients of expansion in a hierarchical basis; and by using the aforementioned fast transform, we construct two very efficient sparse spectral-Galerkin methods for a model elliptic equation. In particular, the Chebyshev–Legendre–Galerkin method leads to a sparse matrix with a much lower number of nonzero elements than that of low-order sparse grid methods based on finite elements or wavelets, and can be efficiently solved by a suitable sparse solver. Ample numerical results are presented to demonstrate the efficiency and accuracy of our algorithms.