A Discrete Domain Decomposition Method for Acoustics with Uniform Exponential Rate of Convergence Using Non-local Impedance Operators

Claeys, Xavier; Collino, Francis; Joly, Patrick; Parolin, Emile

doi:10.1007/978-3-030-56750-7_35

Cited by 5 publications

(7 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Another alternative, advocated in [12,13,27], is to use suitable non-local operators, realized in practice with integral operators, as impedance operators. One of the strengths of this later approach is to rely on a solid theoretical basis that systematically guarantees (huniform [9]) geometric convergence, provided that certain properties of injectivity, surjectivity and positivity (in suitable trace spaces) are satisfied by the impedance operator.…”

Section: Introductionmentioning

confidence: 99%

Robust treatment of cross points in Optimized Schwarz Methods

Claeys¹,

Parolin²

2020

Preprint

Self Cite

View full text Add to dashboard Cite

In the field of Domain Decomposition (DD), Optimized Schwarz Method (OSM) appears to be one of the prominent techniques to solve large scale time-harmonic wave propagation problems. It is based on appropriate transmission conditions using carefully designed impedance operators to exchange information between sub-domains. The efficiency of such methods is however hindered by the presence of cross-points, where more than two sub-domains abut, if no appropriate treatment is provided.In this work, we propose a new treatment of the cross-point issue for the Helmholtz equation that remains valid in any geometrical interface configuration. We exploit the multi-trace formalism to define a new exchange operator with suitable continuity and isometry properties. We then develop a complete theoretical framework that generalizes classical OSM to partitions with cross points and contains a rigorous proof of geometric convergence, uniform with respect to the mesh discretization, for appropriate positive impedance operators. Extensive numerical results in 2D and 3D are provided as an illustration of the performance of the proposed method.

show abstract

Section: Introductionmentioning

confidence: 99%

Robust treatment of cross points in Optimized Schwarz Methods

Claeys¹,

Parolin²

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In the following, we equip the Hilbert space V with the (T-dependent) norm naturally inherited from the H −s -norm defined by ( 5), that we still denote • for simplicity. From (8), it is clear that the operators Π and S are continuous in V. Obviously, Π is an isometry while, from the identity (6) (applied in Ω 1 and Ω 2 ), we immediately infer that, for any…”

Section: Convergence Analysismentioning

confidence: 93%

Nonoverlapping Domain Decomposition Methods for Time Harmonic Wave Problems

Claeys

Collino

Joly

et al. 2022

Domain Decomposition Methods in Science and Engineering XXVI

View full text Add to dashboard Cite

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

show abstract

“…Collino, Ghanemi and Joly (2000) showed that for a decomposition without cross-points, the convergence of the relaxed Schwarz iteration can be geometric if the Taylor of order zero transmission ∂ n + iω is enhanced by a square-root operator to ∂ n + iω α − βω −2 ∂ yy . Claeys et al (2020) analysed the discrete version and showed that the convergence rate is uniform in the mesh size. For recent progress along the direction of non-local transmission conditions, see Lecouvez, Stupfel, Joly and Collino (2014), Collino, Joly and Lecouvez (2020), Parolin (2020), Claeys and Parolin (2021) and Claeys (2021).…”

Section: Parallel Schwarz Methods For the Free-space Wave Problemmentioning

confidence: 99%

“…iterations, independently of the number of subdomains, the decomposition and the partial differential equation that is solved. Such a global communication component is also present in the recent work by Claeys, Collino, Joly and Parolin (2020) and Claeys and Parolin (2021) for time harmonic wave propagation problems, which is based on earlier work of Claeys (2019), where the multi-trace formulation was interpreted as an optimized Schwarz method, including cross-points.…”

Section: Introductionmentioning

confidence: 99%

“…iterations, independently of the number of subdomains, the decomposition and the partial differential equation that is solved. Such a global communication component is also present in the recent work (Claeys, Collino, Joly andParolin 2020, Claeys andParolin 2022) for time harmonic wave propagation problems, which is based on earlier work of Claeys (2019), where the multi-trace formulation was interpreted as an optimized Schwarz method, including cross points. We will use in what follows the two specific domain decompositions shown in Figure 1.2, namely one dimensional, sequential or strip decompositions, and two dimensional decompositions including cross points.…”

Section: Contents 1 Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Schwarz methods by domain truncation

Gander¹,

Hui²

2022

Acta Numerica

View full text Add to dashboard Cite

Schwarz methods use a decomposition of the computational domain into subdomains and need to impose boundary conditions on the subdomain boundaries. In domain truncation one restricts the unbounded domain to a bounded computational domain and must also put boundary conditions on the computational domain boundaries. In both fields there are vast bodies of literature and research is very active and ongoing. It turns out to be fruitful to think of the domain decomposition in Schwarz methods as a truncation of the domain onto subdomains. Seminal precursors of this fundamental idea are papers by Hagstrom, Tewarson and Jazcilevich (1988), Després (1990) and Lions (1990). The first truly optimal Schwarz method that converges in a finite number of steps was proposed by Nataf (1993), 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.A key application of Schwarz methods with such sophisticated transmission conditions turned out to be time-harmonic wave propagation problems, because classical Schwarz methods simply do not work in this case. The past decade has given us many new Schwarz methods based on domain truncation. One review from an algorithmic perspective (Gander and Zhang 2019) showed the equivalence of many of these new methods to optimized Schwarz methods. The analysis of optimized Schwarz methods, however, is lagging behind their algorithmic development. The general abstract Schwarz framework cannot be used for the analysis of these methods, and thus there are many open theoretical questions about their convergence. Just as for practical multigrid methods, Fourier analysis has been instrumental for understanding the convergence of optimized Schwarz methods and for tuning their transmission conditions. Similar to local Fourier mode analysis in multigrid, the unbounded two-subdomain case is used as a model for Fourier analysis of optimized Schwarz methods due to its simplicity. Many aspects of the actual situation, e.g. boundary conditions of the original problem and the number of subdomains, were thus neglected in the unbounded two-subdomain analysis. While this gave important insight, new phenomena beyond the unbounded two-subdomain models were discovered.This present situation is the motivation for our survey: to give a comprehensive review and precise exploration of convergence behaviours of optimized Schwarz methods based on Fourier analysis, taking into account the original boundary conditions, many-subdomain decompositions and layered media. We consider as our model problem the operator $-\Delta + \eta $ in the diffusive case $\eta>0$ (screened Laplace equation) or the oscillatory case $\eta <0$ (Helmholtz equation), in order to show the fundamental difference in behaviour of Schwarz solvers for these problems. The transmission conditions we study include the lowest-order absorbing conditions (Robin), and also more advanced perfectly matched layers (PMLs), both developed first for domain truncation. Our intensive work over the last two years on this review has led to several new results presented here for the first time: in the bounded two-subdomain analysis for the Helmholtz equation, we see strong influence of the original boundary conditions imposed on the global problem on the convergence factor of the Schwarz methods, and the asymptotic convergence factors with small overlap can differ from the unbounded two-subdomain analysis. In the many-subdomain analysis, we find the scaling with the number of subdomains, e.g. when the subdomain size is fixed, robust convergence of the double-sweep Schwarz method for the free-space wave problem, either with fixed overlap and zeroth-order Taylor conditions or with a logarithmically growing PML, and we find that Schwarz methods with PMLs work like smoothers that converge faster for higher Fourier frequencies; in particular, for the free-space wave problem, plane waves (in the error) passing through interfaces at a right angle converge more slowly. In addition to our main focus on analysis in Sections 2 and 3, we start in Section 1 with an expository historical introduction to Schwarz methods, and in Section 4 we give a brief interpretation of the recently proposed optimal Schwarz methods for decompositions with cross-points from the viewpoint of transmission conditions. We conclude in Section 5 with a summary of open research problems. In Appendix A we provide a Matlab program for a block LU form of an optimal Schwarz method with cross-points, and in Appendix B we give the Maple program for the two-subdomain Fourier analysis.

show abstract

A Discrete Domain Decomposition Method for Acoustics with Uniform Exponential Rate of Convergence Using Non-local Impedance Operators

Cited by 5 publications

References 0 publications

Robust treatment of cross points in Optimized Schwarz Methods

Robust treatment of cross points in Optimized Schwarz Methods

Nonoverlapping Domain Decomposition Methods for Time Harmonic Wave Problems

Schwarz methods by domain truncation

Contact Info

Product

Resources

About