On the Spectrum of Deflated Matrices with Applications to the Deflated Shifted Laplace Preconditioner for the Helmholtz Equation

Robust Superlinear Krylov Convergence for Complex Noncoercive Compact-Equivalent Operator Preconditioners

“…where \varepsi > 0 is a properly chosen ``absorption"" parameter, and \mu > 0 is a suitably chosen constant; see, e.g., [12,13,16,18,20,37] in the context of multigrid solvers and [18,28,39] involving domain decomposition methods. The behavior of such preconditioners and choices of shifting parameters have also been analyzed [15,16,19]; typically \varepsi = O(\kappa 2 ) or \varepsi = O(\kappa ) are used.…”

“…Then section 5 provides some applications in H 1 spaces. Here detailed attention is paid to interior Helmholtz equations and shifted Laplace preconditioners, to which a lot of recent research has been devoted (e.g., [13,25,20,37]), however only focused on the aspects of linear convergence. We also indicate other applications and further directions.…”

On the convergence of two‐level Krylov methods for singular symmetric systems

“…For instance, in the work of Erlangga et al, M −1 represents an action of multigrid preconditioner for the Helmholtz matrix, which acts on the shifted Laplace (or modified Helmholtz) matrix. Theoretical results for the MK method (two‐level) applied to the Helmholtz equation can be found in the works of García Ramos et al and Erlangga et al Furthermore, the inclusion of such a matrix is done in MK via a modification of the original matrix A to , , or , where M 1 M 2 = M . For the Galerkin‐type approach, this modification leads to a coarse‐grid structure, which is different from that in standard multigrid/level methods.…”