De ation in preconditioned conjugate gradient methods for Finite Element Problems

We investigate the in uence of the value of de ation vectors at interfaces on the rate of convergence of preconditioned conjugate gradient methods applied to a Finite Element discretization for an elliptic equation. Our set-up is a Poisson problem in two dimensions with continuous or discontinuous coecients that vary in several orders of magnitude. In the continuous case we are interested in the convergence acceleration of de ation on block preconditioners. For the discontinuous case the Finite Element discretization gives a sym-metric matrix with very large condition number and hence many iterations are needed to obtain a solution using conjugate gradients. We use an incomplete Choleski decomposition for the symmetric discretization matrix. Subsequently, de ation is applied to eliminate the disadvantageous eects to convergence caused by the remaining small eigenvalues. Here we investigate the eects of several variants of de ation and we propose an optimal choice for the de ation vectors. Finally, we apply de ation to a parallel preconditioned conjugate method to accellerate the calculation. Mathematics Classication: 65F10 (Iterative methods for linear systems), 76S05 (Flow in Porous Media