INCOMPLETE BLOCK FACTORIZATION PRECONDITIONING FOR LINEAR-SYSTEMS ARISING IN THE NUMERICAL-SOLUTION OF THE HELMHOLTZ-EQUATION

The application of the finite difference method to discretize the comp lex Helmholtz equation on a bounded region in the plane produces a lin ear system whose coefficient matrix is block tridiagonal and is some ( complex) perturbation of an M-matrix. The matrix is also complex symme tric, and its real part is frequently indefinite. Conjugate gradient t ype methods are available for this kind of linear systems, but the pro blem of choosing a good preconditioner remains. We first establish two existence results for incomplete block factorizations of matrices (of special type). In the case of the complex Helmholtz equation, specifi c incomplete block factorization exists for the resulting complex matr ix and its real part if the mesh size is reasonably small. Numerical e xperiments show that using these two incomplete block factorizations a s preconditioners can give considerably better convergence results tha n simply using a preconditioner that is good for the Laplacian also as a preconditioner for the complex system. The latter idea has been use d by many authors for the real case.