Analysis of parallel Schwarz algorithms for time-harmonic problems using block Toeplitz matrices

Bootland, Niall; Dolean, Victorita; Kyriakis, Alexandros; Pestana, Jennifer

doi:10.1553/etna_vol55s112

Cited by 9 publications

(9 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A short remark about the non-scalability in case of external Neumann conditions is given in [3]. Similar results have been recently presented in [1] for time-harmonic problems. The goal of this work is to study the effect of different (possibly mixed) external boundary conditions on convergence and scalability of PSM and OSM.…”

Section: Introductionsupporting

confidence: 69%

“…This work is concerned with convergence and weak scalability 1 analysis of one-level parallel Schwarz method (PSM) and optimized Schwarz method (OSM) for the solution of the problem…”

Section: Introductionmentioning

confidence: 99%

“…However, the recent work [2], published in the field of molecular dynamics, has drawn attention to the opposite case in which the number of subdomains increases, but their size remains unchanged, and, as a result, the size of the whole domain Ω increases. In this setting, weak scalability of PSM and OSM for (1) with Dirichlet boundary conditions is studied in [4,3]. Scalability results for the PSM in case of more general geometries of the (sub)domains are presented in [5,6,7].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the effect of boundary conditions on the scalability of Schwarz methods

Ciaramella¹,

Mechelli²

2021

Preprint

View full text Add to dashboard Cite

Section: Introductionsupporting

confidence: 69%

“…This work is concerned with convergence and weak scalability 1 analysis of one-level parallel Schwarz method (PSM) and optimized Schwarz method (OSM) for the solution of the problem…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the effect of boundary conditions on the scalability of Schwarz methods

Ciaramella¹,

Mechelli²

2021

Preprint

View full text Add to dashboard Cite

“…Theoretical work to understand the effectiveness of this approach can be found in [35,37] and for the heterogeneous problem in [34]; see also [33]. To be scalable, such additive Schwarz methods require a second level, known as a coarse space (though see [9] for a novel analysis for the absorptive problem). For the twolevel methods in [35,36] this is provided through a coarse grid; an approach which is effective also for the time-harmonic Maxwell problem [4].…”

mentioning

confidence: 99%

Can DtN and GenEO coarse spaces be sufficiently robust for heterogeneous Helmholtz problems?

Bootland,

Dolean

2022

Preprint

Self Cite

View full text Add to dashboard Cite

Numerical solution of heterogeneous Helmholtz problems presents various computational challenges, with descriptive theory remaining out of reach for many popular approaches. Robustness and scalability are key for practical and reliable solvers in large-scale applications, especially for large wave number problems. In this work we explore the use of a GenEO-type coarse space to build a two-level additive Schwarz method applicable to highly indefinite Helmholtz problems. Through a range of numerical tests on a 2D model problem, discretised by finite elements on pollution-free meshes, we observe robust, wave number independent convergence and scalability of our approach. We further provide results showing a favourable comparison with the DtN coarse space. Our numerical study shows promise that our solver methodology can be effective for challenging heterogeneous applications.

show abstract

“…Traditionally, such transmission conditions are derived and optimized for simple two subdomain configurations, like in optimized Schwarz methods, see [6,17,3,19,11,21,20,22,1,16,15] and references therein. We investigate here the optimization of transmission conditions directly for the many subdomain case, and also study the optimization problem in the limit when the number of subdomains goes to infinity, using the tool of limiting spectra, see [2] and the references therein. Our analysis for overlapping methods shows that the optimized transmission conditions in the many subdomain case behave asymptotically when the overlap goes to zero like the optimized transmission conditions for the two subdomain case, only the constants, which we derive in closed form for a given number of subdomains, differ.…”

mentioning

confidence: 99%

Closed form optimized transmission conditions for complex diffusion with many subdomains

Dolean¹,

Gander²,

Kyriakis³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

Optimized transmission conditions in domain decomposition methods have been the focus of intensive research efforts over the past decade. Traditionally, transmission conditions are optimized for two subdomain model configurations, and then used in practice for many subdomains. We optimize here transmission conditions for the first time directly for many subdomains for a class of complex diffusion problems. Our asymptotic analysis leads to closed form optimized transmission conditions for many subdomains, and shows that the asymptotic best choice in the mesh size only differs from the two subdomain best choice in the constants, for which we derive the dependence on the number of subdomains explicitly, including the limiting case of an infinite number of subdomains, leading to new insight into scalability. Our results include both Robin and Ventcell transmission conditions, and we also optimize for the first time a two-sided Ventcell condition. We illustrate our results with numerical experiments, both for situations covered by our analysis and situations that go beyond.

show abstract

Analysis of parallel Schwarz algorithms for time-harmonic problems using block Toeplitz matrices

Cited by 9 publications

References 29 publications

On the effect of boundary conditions on the scalability of Schwarz methods

On the effect of boundary conditions on the scalability of Schwarz methods

Can DtN and GenEO coarse spaces be sufficiently robust for heterogeneous Helmholtz problems?

Closed form optimized transmission conditions for complex diffusion with many subdomains

Contact Info

Product

Resources

About