The rate of convergence of iterations of implicit incomplete factorization methods (Q1309051)

Implicit incomplete factorization methods for solving five-point mesh equations, which arise in finite difference and finite element approximations of two-dimensional boundary value problems, are considered. For the final linear system \(Au= f\) it is assumed that \(A\) represents a specially structured \(M\)-matrix and for its solution a conjugate gradient method with a skillfully selected preconditioning matrix is proposed. This method can be considered as a generalization of the block symmetric successive overrelaxation method (BSSOR). Under the assumption that \(A\) is a symmetric positive definite \(M\)-matrix (Stieltjes matrix) the convergence rate is estimated. The estimation for a fixed relaxation parameter is (for a model problem) worse than the result for the optimal BSSOR. Two numerical examples show the efficiency of the method in comparison with the optimal BSSOR.