Asymptotic Convergence of a New Spectral Method for the Solution in Time of Finite Element Flow Equations

An eecient method (DACG, Deeated Accelerated Conjugate Gradient) has recently been developed, based on the conjugate gradient (CG) minimization of the Rayleigh quotient over successive deeated subspaces of decreasing size. A new version of this algorithm is presented and discussed to evaluate the 40 leftmost eigenpairs of FE problems of increasing size up to 10500. Its asymptotic convergence rate is shown to be inversely proportional to the square root of the spectral condition number of the Hessian of the Rayleigh quotient in the current restricted subspace. Numerical results show that preconditioning the CG method with the inverse of the incomplete factors greatly improves convergence which may prove to be slow only toward very close eigenvalues.