Optimized Schwarz Methods with Overlap for the Helmholtz Equation

Gander, Martin J.; Zhang, Hui

doi:10.1007/978-3-319-05789-7_17

Cited by 19 publications

(23 citation statements)

References 11 publications

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“…For Helmholtz problems, the second-order Taylor expansion was used in [43], square-root based nonlocal conditions were studied in [35,86], best approximations of zero-and second-order were sought in [31,69,76,83,84,158], Padé approximants with complexshifted wavenumbers were used in [15], PMLs were first employed in [146,161], and recently some rational interpolants were tested in [109] for waveguide problems. For a numerical comparison of low-order and high-order transmission conditions for the overlapping Schwarz methods, we refer to [84]. Parallel to the development of optimized Schwarz methods, absorbing transmission conditions have also found use in the analytic incomplete LU (AILU) preconditioner; see [77,78].…”

Section: Direct and Iterative Solvers After Discretization Of Equationmentioning

confidence: 99%

A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz Methods

Gander¹,

Zhang²

2019

SIAM Rev.

Self Cite

149

114

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Solving time-harmonic wave propagation problems by iterative methods is a difficult task, and over the last two decades, an important research effort has gone into developing preconditioners for the simplest representative of such wave propagation problems, the Helmholtz equation. A specific class of these new preconditioners are considered here. They were developed by researchers with various backgrounds using formulations and notations that are very different, and all are among the most promising preconditioners for the Helmholtz equation.The goal of the present manuscript is to show that this class of preconditioners are based on a common mathematical principle, and they can all be formulated in the context of domain decomposition methods called optimized Schwarz methods. This common formulation allows us to explain in detail how and why all these methods work. The domain decomposition formulation also allows us to avoid technicalities in the implementation description we give of these recent methods.The equivalence of these methods with optimized Schwarz methods translates at the discrete level into equivalence with approximate block LU decomposition preconditioners, and we give in each case the algebraic version, including a detailed description of the approximations used. While we chose to use the Helmholtz equation for which these methods were developed, our notation is completely general and the algorithms we give are written for an arbitrary second order elliptic operator. The algebraic versions are even more general, assuming only a connectivity pattern in the discretization matrix.All these new methods studied here are based on sequential decompositions of the problem in space into a sequence of subproblems, and they have in their optimal form the property to lead to nilpotent iterations, like an exact block LU factorization. Using our domain decomposition formulation, we finally present an algorithm for two dimensional decompositions, i.e. decompositions that contain cross points, which is still nilpotent in its optimal form. Its approximation is currently an active area of research, and it would have been difficult to discover such an algorithm without the domain decomposition framework. 1 We use this simplest form of the Helmholtz equation only here at the beginning, and treat in the main part the more complete formulation given in Equation (11). 2 We assume here homogeneous Dirichlet boundary conditions and well-posedness for simplicity at the beginning, see Section 4 for more information.The overlap has no influence on the two step convergence property of the optimal parallel Schwarz method 5 . With J subdomains, as indicated in Figure 1 on the right, 4 The right hand side on the interface is in fact an exact or transparent boundary condition for the neighboring subdomain.5 This will be different if one uses approximations of the DtN operators, as we will see. * 10 we use Matlab notation for concatenating column vectors vertically to avoid having to use the transpose symbol T .

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Section: Direct and Iterative Solvers After Discretization Of Equationmentioning

confidence: 99%

A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz Methods

Gander¹,

Zhang²

2019

SIAM Rev.

Self Cite

149

114

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“…, and u + = u dif + e ıβ k 0 (z−a) U + k0 , we get a solution of (11) + (13) which does not satisfy compatibility relations (12). …”

Section: Resultsmentioning

confidence: 99%

“…We prove here that if ω is an eigenfrequency of the homogeneous problem defined by (13) + (18), then the limit problem (11) + (13) admits solutions that do not satisfy the compatibility relations (12). We want to find (u − , u b , u + ) solution of (11) as incident field.…”

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Iterative methods for scattering problems in isotropic or anisotropic elastic waveguides

et al. 2016

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We consider the time-harmonic problem of the diffraction of an incident propagative mode by a localized defect, in an infinite elastic waveguide. We propose several iterative algorithms to compute an approximate solution of the problem, using a classical finite element discretization in a small area around the perturbation, and a modal expansion in the unbounded straight parts of the guide. Each algorithm can be related to a socalled domain decomposition method, with an overlap between the domains. Specific transmission conditions are used, so that at each step of the algorithm only the sparse finite element matrix has to be inverted, the modal expansion being obtained by a simple projection, using a bi-orthogonality relation. The benefit of using an overlap between the finite element domain and the modal domain is emphasized. An original choice of transmission conditions is proposed which enhances the effect of the overlap and allows us to handle arbitrary anisotropic materials. As a by-product, we derive transparent boundary conditions for an arbitrary anisotropic waveguide. The transparency of these new boundary conditions is checked for twoand three-dimensional anisotropic waveguides. Finally, in the isotropic case, numerical validation for twoand three-dimensional waveguides illustrates the efficiency of the new approach, compared to other existing methods, in terms of number of iterations and CPU time.

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“…The work of Japhet was originally carried on the advection-diffusion equation in the plane, without overlap, and using second order transmission conditions. Optimized Schwarz methods are now well-studied for symmetric partial differential equations, for example for the Laplace and modified Helmholtz equations (see [GHN01,Gan03] and references therein) and the Helmholtz equation (see [Gan01,GMN02]). …”

Section: Introductionmentioning

confidence: 99%

Optimized Schwarz Methods with Robin Conditions for the Advection-Diffusion Equation

Dubois

Lecture Notes in Computational Science and Engineering

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Summary. We study optimized Schwarz methods for the stationary advectiondiffusion equation in two dimensions. We look at simple Robin transmission conditions, with one free parameter. In the nonoverlapping case, we solve exactly the associated min-max problem to get a direct formula for the optimized parameter. In the overlapping situation, we solve only an approximate min-max problem. The asymptotic performance of the resulting methods, for small mesh sizes, is derived. Numerical experiments illustrate the improved convergence compared to other Robin conditions.

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Optimized Schwarz Methods with Overlap for the Helmholtz Equation

Cited by 19 publications

References 11 publications

A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz Methods

A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz Methods

Iterative methods for scattering problems in isotropic or anisotropic elastic waveguides

Optimized Schwarz Methods with Robin Conditions for the Advection-Diffusion Equation

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