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 complex Helmholtz equation on a bounded region in the plane produces a linear system whose coefficient matrix is block tridiagonal and is some (complex) perturbation of an M-matrix. The matrix is also complex symmetric, and its real part is frequently indefinite. Conjugate gradient type methods are available for this kind of linear systems, but the problem 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, specific incomplete block factorization exists for the resulting complex matrix and its real part if the mesh size is reasonably small. Numerical experiments show that using these two incomplete block factorizations as preconditioners can give considerably better convergence results than simply using a preconditioner that is good for the Laplacian also as a preconditioner for the complex system. The latter idea has been used by many authors for the real case.