Deflation in Preconditioned Conjugate Gradient Methods for Finite Element Problems

We investigate the influence of the value of deflation 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 coefficients that vary in several orders of magnitude. In the continuous case we are interested in the convergence acceleration of deflation on block preconditioners. For the discontinuous case the Finite Element discretization gives a symmetric 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, deflation is applied to eliminate the disadvantageous effects to convergence caused by the remaining small eigenvalues. Here we investigate the effects of several variants of deflation and we propose an optimal choice for the deflation vectors. Finally, we apply deflation to a parallel preconditioned conjugate method to accellerate the calculation.

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