ADAPTIVE MULTIGRID TECHNIQUES FOR LARGE-SCALE EIGENVALUE PROBLEMS - SOLUTIONS OF THE SCHRODINGER-PROBLEM IN 2 AND 3 DIMENSIONS

Multigrid (MG) algorithms for large-scale eigenvalue problems (EP), obtained from discretizations of partial differential EP, have often been shown to be more efficient than single level eigenvalue algorithms. This paper describes a robust and efficient, adaptive MG eigenvalue algorithm. The robustness of the present approach is a result of a combination of MG techniques introduced here, i.e., the completion of clusters; the adaptive treatment of clusters; the simultaneous treatment of solutions in each cluster; the miltigrid projection (MGP) coupled with backrotations; and robustness tests. Due to the MGP, the algorithm achieves a better computational complexity and better convergence rates than previous MG eigenvalue algorithms that use only fine level projections. These techniques overcome major computational difficulties related to equal and closely clustered eigenvalues. Some of these difficulties were not treated in previous MG algorithms. Computational examples for the Schrödinger eigenvalue problem in two and three dimensions are demonstrated for cases of special computational difficulties, which are due to equal and closely clustered eigenvalues. For these cases, the algorithm requires O(qN) operations for the calculation of q eigenvectors of size N, using a second order approximation. The total computational cost is equivalent to only a few Gause-Seidel relaxations per eigenvector. © 1995 The American Physical Society.