SHEM: An Optimal Coarse Space for RAS and Its Multiscale Approximation

Gander, Martin J.; Loneland, Atle

doi:10.1007/978-3-319-52389-7_32

Cited by 24 publications

(34 citation statements)

References 15 publications

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“…While there is currently no practical realization of this algorithm, the fact that the algorithm converges in two iterations, independently of the number of subdomains, suggests that a coarse space component is active in this optimal algorithm. Coarse spaces leading to nilpotent iterations have first been described in the lecture notes [68], and then in [70,71], with successful approximations in [74,75]. The property of domain decomposition methods in general to be nilpotent has only very recently been investigated in more detail, see [24].…”

Section: Discussionmentioning

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.

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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: Discussionmentioning

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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“…Notice that the dimension of V N c is proportional to N . It is clear that the coarse space V N c has mainly information condensed on the boundaries of the objects, similar to the Spectral Harmonically Enriched Multiscale coarse space SHEM in domain decomposition [21,20], which contains mainly information on the interfaces between subdomains, see also [19,18]. It is important to remark that the construction of each function ϕ j,n would require the solution of problem (6.1), which requires the same computational effort of the original problem (2.3).…”

Section: Scalability Analysis and Coarse Correctionmentioning

confidence: 99%

Methods of Reflections: relations with Schwarz methods and classical stationary iterations, scalability and preconditioning.

Ciaramella¹,

Gander²,

Halpern³

et al. 2019

The SMAI Journal of Computational Mathematics

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“…The basis functions are obtained by solving local problems mimicking (1) at the scale of mesh elements, with carefully chosen right-hand sides and boundary conditions. The vanilla version of the approach, called linear MsFEM, uses as basis functions the solutions to these local problems, posed on each mesh element, with null right-hand sides and with the coarse P1 elements as Dirichlet boundary conditions (see (12) below for the precise definition of these basis functions). Various improvements of that version are possible.…”

Section: Introductionmentioning

confidence: 99%

An MsFEM approach enriched using Legendre polynomials

Legoll¹,

Rothe²,

Bris³

et al. 2021

Preprint

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We consider a variant of the conventional MsFEM approach with enrichments based on Legendre polynomials, both in the bulk of mesh elements and on their interfaces. A convergence analysis of the approach is presented. Residue-type a posteriori error estimates are also established. Numerical experiments show a significant reduction in the error at a limited additional off-line cost. In particular, the approach developed here is less prone to resonance errors in the regime where the coarse mesh size H is of the order of the small scale ε of the oscillations.

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SHEM: An Optimal Coarse Space for RAS and Its Multiscale Approximation

Cited by 24 publications

References 15 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

Methods of Reflections: relations with Schwarz methods and classical stationary iterations, scalability and preconditioning.

An MsFEM approach enriched using Legendre polynomials

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