Accelerated simultaneous iterations for large finite element eigenproblems

An accelerated simultaneous iteration method is presented for the solution of the generalized eigenproblem Ax = λBx, where A and B are real sparse symmetric positive definite matrices. The approach is well suited for the determination of the leftmost eigenpairs of problems with large size N. The procedure relies on the optimization of the Rayleigh quotient over a subspace of orthogonal vectors by a conjugate gradient technique effectively preconditioned with the pointwise incomplete Cholesky factorization. The method is applied to the evaluation of the smallest 15 eigenpairs of finite element models with size ranging between 150 and 2300. The numerical experiments show that, while the simultaneous conjugate gradient scheme fails to converge, the accelerated iterations yield accurate results in a number of steps which is much smaller than N. The new approach does not require the a priori estimate of any empirical parameter and appears to be a robust, reliable, and efficient tool for the partial eigensolution of large finite element problems. © 1989