The influence of deflation vectors at interfaces on the deflated conjugate gradient method

We investigate the influence of the value of deflation vectors at interfaces on the rate of convergence of preconditioned conjugate gradient methods. Our set-up is a Laplace problem in two dimensions with continuous or discontinuous coeffcients that vary in several orders of magnitude. In the continuous case we are interested in the convergence acceleration of deflation on block preconditioners. The finite volume discretization gives a matrix with very distinct eigenvalues and hence many iterations are needed to obtain a solution using conjugate gradients. We use an incomplete Choleski preconditioning for the symmetric discretization matrix. Subsequently, deflation is applied to eliminate the disadvantageous effects to convergence caused by the remaining small eigenvalues. Here we show the effects of several variants of algebraic deflation and we propose an optimal choice for the deflation vectors.Electrical Engineering, Mathematics and Computer Scienc