Incomplete factorization-based preconditionings for solving the Helmholtz equation

Comparison of multigrid and incomplete LU shifted-Laplace preconditioners for the inhomogeneous Helmholtz equation

“…In the following subsections, we will shortly discuss some preconditioners for the Helmholtz equation, namely the shifted-Laplace preconditioner (our method of choice), the incomplete factorization of a "modified" Helmholtz operator [9] and the separation-of-variables preconditioner [11] (with which we compare our numerical results).…”

“…2 [9] An ILU factorization may not be stable if A is not an M-matrix, which is the case for the discrete Helmholtz equation. In [9] approximations of A are proposed so that ILU factorizations can be I , a constraint is set so that the preconditioned system AM −1 I is definite or "less indefinite".…”

“…We show that, based on the GMRES convergence theory, the bound of the convergence rate can be obtained if radiation condition (2) is used. Finally, we compare the overall performance of the shifted-Laplace preconditioner with other preconditioners (i.e., separation-of-variables [11] and incomplete factorization [9]). …”

A damping preconditioner for time-harmonic wave equations in fluid and elastic material

“…The performance of the damped preconditioner was compared to a modified incomplete Cholesky factorization (MIC) preconditioner [27]. The algorithm presented in [40] is used for the MIC(l) approximation of A −1 , where the parameter l describes the level of fillin in the factorization.…”

“…For acoustic and elastic problems in homogenous medium, domain imbedding/fictitious domain methods in [23,24,25,26] have been fairly effective, but these methods are pretty restrictive and not well-suited in general, complicated domains. An incomplete factorization preconditioner has been considered in [27], for example, and in [28] a tensor product preconditioner is used.…”