A new fast method of solving the high dimensional elliptic eigenvalue problem

Abstract

In this paper, we develop a novel method to solve the elliptic eigenvalue problem. The univariate multi-wavelet approach provides a simple diagonal preconditioner for second order elliptic problems, which gives an almost constant condition number for efficiently solving the corresponding linear system. Here, we shall consider a new fast numerical approach for approximating the smallest elliptic eigenvalue by using the multi-wavelet basis in the multi-grid discretization scheme. Moreover, we develop a new numerical scheme coupled with sparse grids method in the calculation. This new approach saves storage in degrees of freedom and thus is more efficient in the computation. Several numerical experiments are provided for validating the proposed numerical scheme, which show that our method retains the optimal convergence rate for the smallest eigenvalue approximation with much less computational cost comparing with ’eigs’ in full grids.