Department of Applied Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot, Israel

Shlomo Ta'asan

Department of Applied Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot, Israel

and ICASE, NASA Langley Research Center, Hampton, Va 23665, USA

Abstract

A multilevel adaptive algorithm to compute a few eigenvectors and the corresponding eigenvalues for large scale eigenvalue problems, is presented. The approach is to simultaneously separate the sought eigenspaces and the eigenvectors within the eigenspaces using relaxations, projections and multilevel transfers. There are previous multilevel eigenvalue algorithms where the major amount of work is associated with the separation of eigenvectors by fine level work. In the algorithm presented here, this task is performed by coarse level separation techniques, resulting in large savings. The introduced separation techniques are a multilevel projection (a Generalysed Rayleigh Ritz projection) and the backrotations. The treatment of equal and close clustered eigenvalues is specially emphasized. The algorithm is robust due to 1) an adaptive treatment of clusters on different levels and 2) incorporated robustness tests. Computational examples are presented where Schr\"odinger type eigenvalue problems in 2-D and 3-D are solved with the efficiency of the Poisson multigrid solver. Solutions for a second order scheme are obtained by 1-FMG-V(1,1) in O(qN) work, for q eigenvectors of size N on the finest level. The separation accuracy is presented for equal and close clustered eigenvalues.