A two‐level domain decomposition method with accurate interface conditions for the Helmholtz problem

Abstract

Summary A new and efficient two‐level, non‐overlapping domain decomposition (DD) method is developed for the Helmholtz equation in the two Lagrange multiplier framework. The transmission conditions are designed by utilizing perfectly matched discrete layers (PMDLs), which are a more accurate representation of the exterior Dirichlet‐to‐Neumann map than the polynomial approximations used in the optimized Schwarz method. Another important ingredient affecting the convergence of a DD method, namely, the coarse space augmentation, is also revisited. Specifically, the widely successful approach based on plane waves is modified to that based on interface waves, defined directly on the subdomain boundaries, hence ensuring linear independence and facilitating the estimation of the optimal size for the coarse problem. The effectiveness of both PMDL‐based transmission conditions and interface‐wave‐based coarse space augmentation is illustrated with an array of numerical experiments that include comprehensive scalability studies with respect to frequency, mesh size and the number of subdomains. Copyright © 2015 John Wiley & Sons, Ltd.

A non-overlapping domain decomposition method with high-order transmission conditions and cross-point treatment for Helmholtz problems

“…The problem finally consists in solving the main field u on the rectangular domain with a HABC on each edge by (4). Auxiliary fields defined on the edges are governed by 1D Helmholtz equations through (6) and are coupled at the corners by auxiliary relations by ( 7)-( 8) and auxiliary variables using (9).…”

“…In parallel, FETI methods were adapted to Helmholtz problems as FETI-H [27,37] and FETI-DPH [39] techniques, which can be interpreted as substructuring DDMs with optimized transmission conditions and preconditioning techniques. Later, domain decomposition strategies with HABC-based transmission conditions were developed to improve the convergence rate and robustness of the methods [13,14,52,57], as well as PML-based approaches [4,68,69,76] and non-local transmission conditions [24,54,71]. As for ABCs, transmission boundary conditions related to HABCs and PML represent a good compromise between the basic impedance conditions (which lead to suboptimal convergence) and the exact Dirichlet-to-Neumann (DtN) map related to the complementary of the subdomain (which is expensive to compute).…”

“…Recently, cross-point treatments have been investigated for second-order transmission operators [29,64] and non-local transmission approaches [21,22]. PMLtype operators have been tested in configurations with cross-points, but only in the context of DDM preconditioning [4,55,68]. The treatment of interior cross-points for optimized solvers with HABCbased, PML-based and non-local transmission conditions is a complicated problem which has, to the best of our knowledge, not been carefully addressed.…”

Corner treatments for high-order local absorbing boundary conditions in high-frequency acoustic scattering

“…Unfortunately, because of the highly oscillatory nature of the wave fields, these solvers lead to discretizations with a large number of unknowns and require the solution of large poorly-conditioned linear systems. Research on accurate and computationally efficient methods is very active: we can mention for instance recent works on high-frequency boundary element methods [7,21,22], high-order finite element methods [14,16,57,68] and domain decomposition methods [9,10,34,77].…”

The method of polarized traces for the 3D Helmholtz equation

“…Engquist and Zhao (1998) introduced absorbing boundary conditions for domain decomposition schemes for elliptic problems and the first application of such techniques to the Helmholtz problem traces back to the AILU factorization (Gander and Nataf (2000)). The sweeping preconditioner, introduced in Engquist and Ying (2011a,b), was † For a review on classical Schwarz methods see (Chan and Mathew, 1994;Toselli and Windlund, 2005); and for other applications of domain decomposition methods for the Helmholtz equations, see (de La Bourdonnaye et al, 1998;Ghanemi, 1998;McInnes et al, 1998;Collino et al, 2000;Magoules et al, 2000;Boubendir, 2007;Astaneh and Guddati, 2016).…”