Gradient Eigenanalysis Onnested Finite

The eecient computation of the leftmost eigenpairs of the generalized symmetric eigenproblem Ax = Bx by a de-ation accelerated conjugate gradient (DACG) method may be enhanced by an improved estimate of the initial eigenvectors obtained with a multigrid (MG) type approach. The DACG algorithm essentially optimizes the Rayleigh quotient in subspaces of decreasing size B-orthogonal to the eigenvectors previously computed by a preconditioned conjugate gradient (CG) scheme. The DACG asymptotic rate of convergence may be shown to be controlled by the relative separation of the eigenvalue being currently sought and the next higher one and can be eeectively accelerated by the use of various precondi-tioners taken from the family of the incomplete Cholesky decompositions of A. The initial rate may be ameliorated by providing an initial guess calculated on nested nite element (FE) grids of growing resolution. The overall algorithm has been applied to structural eigenproblems deened over four nested FE grids. The results for the computation of the 40 smallest eigenpairs indicate that the asymptotic convergence is very much dependent on the actual eigenvalue distribution and may be substantially improved by the use of appropriate and relatively inexpensive preconditioners. The nested iterations (NI) may lead to a marked reduction of the initial iterations on the nest grid level where the solution is nally required. NI decreases the CPU time by a factor of 2.5. The performance of the NI-DACG method is very promising and emphasizes the potential of this new approach in the partial solution of symmetric positive deenite eigen-problems of large and very large size.