Optimized Schwarz Methods without Overlap for the Helmholtz Equation

  title={Optimized Schwarz Methods without Overlap for the Helmholtz Equation},
  author={Martin J. Gander and Fr{\'e}d{\'e}ric Magoul{\`e}s and Fr{\'e}d{\'e}ric Nataf},
  journal={SIAM J. Sci. Comput.},
The classical Schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel. Convergence is achieved through overlap of the subdomains. We study in this paper a variant of the Schwarz method which converges without overlap for the Helmholtz equation. We show that the key ingredients for such an algorithm are the transmission conditions. We derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps… 

Figures and Tables from this paper

Optimized Schwarz Methods for the Advection-Diffusion Equation
The purpose of this work is to compute optimized Robin transmission conditions for the advection-diffusion equation in two dimensions by finding the solution of the min-max problem, and to validate the continuous Fourier analysis performed on a simple model problem only.
Convergence estimates for an optimized Schwarz method for PDEs with discontinuous coefficients
This work derives convergence estimates when the mesh size h along the interface is small and the jump in the coefficient may be large, and improves upon the rate of popular domain decomposition methods such as the Neumann–Neumann method or FETI-DP methods, which was shown to be independent of the jump.
Optimized Schwarz Methods with Overlap for the Helmholtz Equation
This work presents here for the first time a complete analysis of optimized Schwarz methods with overlap for the Helmholtz equation, and obtains closed form asymptotically optimized transmission conditions for the case of two subdomains.
Different local approximations of these optimal absorbing boundary conditions for finite element computations in acoustics are investigated both in the continuous and in the discrete analysis, including high-order optimized continuous absorbing Boundary conditions, and discrete absorbs boundary conditions based on algebraic approximation.
Optimized Schwarz Methods for Maxwell's Equations
It is shown here why the classical Schwarz method applied to both the time harmonic and time discretized Maxwell's equations converges without overlap: the method has the same convergence factor as a simple optimized Schwarz method for a scalar elliptic equation.
On the Relation Between Optimized Schwarz Methods and Source Transfer
Optimized Schwarz methods (OS) use Robin or higher order transmission conditions instead of the classical Dirichlet ones, and here optimality means that the method converges in a finite number of steps.
Optimized Schwarz methods with nonoverlapping circular domain decomposition
This work derives optimized zeroth and second order transmission conditions for a model elliptic operator in two dimensions, and shows why the straight interface analysis results, when properly scaled to include the curvature, are also successful for curved interfaces.
An optimized Schwarz method with two‐sided Robin transmission conditions for the Helmholtz equation
A new optimized Schwarz method without overlap in the 2d case is presented, which uses a different Robin condition for neighbouring subdomains at their common interface, and which is called two‐sided Robin condition.


Symmetrized Method with Optimized Second-Order Conditions for the Helmholtz Equation
A schwarz type domain decomposition method for the Helmholtz equation is considered and the computational domain Ω is decomposed into N nonoverlapping subdomains, increasing the convergence speed dramatically.
Overlapping Schwarz Algorithms for Solving Helmholtz''s Equation
This paper gives a progress report on the development of a new family of domain decomposition methods for the solution of Helmholtz''s equation through a sequence of discontinuous iterates based on overlapping Schwarz methods.
Schwarz Alternating Method
A discrete technique of the Schwarz alternating method is presented, to combine the Ritz-Galerkin and finite element methods, well suited for solving singularity problems in parallel.
The optimized order 2 method : Application to convection-diffusion problems
Domain Decomposition Algorithms for Indefinite Elliptic Problems
This paper presents an additive Schwarz method applied to linear, second order, symmetric or nonsymmetric, indefinite elliptic boundary value problems in two and three dimensions, and shows the rate of convergence to be independent of the number of degrees of freedom and thenumber of local problems.
Is the Pollution Effect of the FEM Avoidable for the Helmholtz Equation Considering High Wave Numbers?
It is proved that, in two and more space dimensions, it is impossible to eliminate the so-called pollution effect of the Galerkin FEM.
The optimized order 2 method: application to convection-diffusion problems
This work presents an iterative, non-overlapping domain decomposition method, which permits to solve very big problems which couldn’t be solved on only one processor, and shows a very fast convergence, nearly independant both of the physical and the discretization parameters.
The Best Interface Conditions for Domain Decomposition Methods : Absorbing Boundary Conditions
We present an iterative domain decomposition method for solving the convection-diffusion equation. In order to have very fast convergence, we use differential interface conditions of order 1 in the
Domain Decomposition Methods in Science and Engineering
Invited Talks.- Non-matching Grids and Lagrange Multipliers.- A FETI Method for a Class of Indefinite or Complex Second- or Fourth-Order Problems.- Hybrid Schwarz-Multigrid Methods for the Spectral
Application of a domain decomposition method with Lagrange multipliers to acoustic problems arising
The purpose of this paper is to study the robustness and efficiency of iterative methods for the solution of the associated interface problem for three-dimensional interior problems arising from the automotive industry.