Modern Solvers for Helmholtz Problems by Domenico Lahaye Jok Tang & Kees Vuik

Author:Domenico Lahaye, Jok Tang & Kees Vuik
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

In Algorithm 2 we present a more detailed pseudocode for the practical MKMG method incorporated in a flexible GMRES method.

Algorithm 1 MKMG as a preconditioner

It is proven in [23] that the eigenvalues of (the ideal MKMG operator) lie on exact the same circles as the CSL-preconditioned linear system for Dirichlet boundary conditions and are enclosed by the same circles for Sommerfeld boundary conditions. Moreover, they are better clustered. This holds for any dimension, any wavenumber and any choice of Z. In [34] exact formulas for the eigenvalues of in the 1D case are given. These theoretical results help to explain the good performance of the MKMG method. Although the eigenvalue formulas show that for k → ∞, the eigenvalues of move to zero, this happens only for very high wavenumbers (see Fig. 6).

Numerical experiments with the ideal MKMG suggests that an h,k-independent fast convergence can be attained. For the practical MKMG method, our numerical results demonstrate a convergence, which is mildly h-dependent and k-dependent (Sect. 5). As h → 0, the convergence can be made practically independent of h. Considering the fast convergence of the ideal MKMG method, improvement from the current practical MKMG can still be achieved by, e.g., having a better approximation of the coarse-grid matrix .

Remark 1

Instead of preconditioning the matrix first and then deflate, alternatively, one can first deflate and then precondition. This is done by the operator

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.