Schwarz methods by domain truncation
arXiv:2207.09791 · doi:10.1017/S0962492922000034
Abstract
Schwarz methods use a decomposition of the computational domain into subdomains and need to put boundary conditions on the subdomain boundaries. In domain truncation one restricts the unbounded domain to a bounded computational domain and also needs to put boundary conditions on the computational domain boundaries. It turns out to be fruitful to think of the domain decomposition in Schwarz methods as truncation of the domain onto subdomains. The first truly optimal Schwarz method that converges in a finite number of steps was proposed in 1994 and used precisely transparent boundary conditions as transmission conditions between subdomains. Approximating these transparent boundary conditions for fast convergence of Schwarz methods led to the development of optimized Schwarz methods -- a name that has become common for Schwarz methods based on domain truncation. Compared to classical Schwarz methods which use simple Dirichlet transmission conditions and have been successfully used in a wide range of applications, optimized Schwarz methods are much less well understood, mainly due to their more sophisticated transmission conditions. This present situation is the motivation for our survey: to give a comprehensive review and precise exploration of convergence behaviors of optimized Schwarz methods based on Fourier analysis taking into account the original boundary conditions, many subdomain decompositions and layered media. The transmission conditions we study include the lowest order absorbing conditions (Robin), and also more advanced perfectly matched layers (PML), both developed first for domain truncation.
arXiv comment32MB, 137 pages, a lot of figures
References in corpus (15)
- A rapidly converging domain decomposition method for the Helmholtz equation
- A parallel sweeping preconditioner for heterogeneous 3D Helmholtz equations
- Domain Decomposition Method for Maxwell's Equations: Scattering off Periodic Structures
- L-Sweeps: A scalable, parallel preconditioner for the high-frequency Helmholtz equation
- An improved sweeping domain decomposition preconditioner for the Helmholtz equation
- A comparison of coarse spaces for Helmholtz problems in the high frequency regime
- An Additive Overlapping Domain Decomposition Method for the Helmholtz Equation
- Analysis of Schwarz methods for a hybridizable discontinuous Galerkin discretization
- Parallel Controllability Methods For the Helmholtz Equation
- Two-level preconditioners for the Helmholtz equation
- A hybrid approach to solve the high-frequency Helmholtz equation with source singularity in smooth heterogeneous media
- Multidirectionnal sweeping preconditioners with non-overlapping checkerboard domain decomposition for Helmholtz problems
- Towards Accuracy and Scalability: Combining Isogeometric Analysis with Deflation to Obtain Scalable Convergence for the Helmholtz Equation
- A hierarchical preconditioner for wave problems in quasilinear complexity
- Inexact subdomain solves using deflated GMRES for Helmholtz problems