Optimized Domain Decomposition Methods for the Spherical Laplacian

Loisel, Sébastien; Côté, Jean; Gander, Martin J.; Laayouni, Lahcen; Qaddouri, Abdessamad

doi:10.1137/080727014

Cited by 15 publications

(15 citation statements)

References 25 publications

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“…This is interesting because the convergence properties of OSM [27] for special domains using Fourier transforms suggest that the condition number of A 2LM varies in the grid parameter h like O(h − 1 2 ). The remainder of the present article will show that this is true in general (cf.…”

Section: Motivation For the 2lm Methodmentioning

confidence: 98%

Condition Number Estimates for the Nonoverlapping Optimized Schwarz Method and the 2-Lagrange Multiplier Method for General Domains and Cross Points

Loisel¹

2013

SIAM J. Numer. Anal.

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The optimized Schwarz method and the closely related 2-Lagrange multiplier method are domain decomposition methods which can be used to parallelize the solution of partial differential equations. Although these methods are known to work well in special cases (e.g., when the domain is a square and the two subdomains are rectangles), the problem has never been systematically stated nor analyzed for general domains with general subdomains. The problem of cross points (when three or more subdomains meet at a single vertex) has been particularly vexing. We introduce a 2-Lagrange multiplier method for domain decompositions with cross points. We estimate the condition number of the iteration and provide an optimized Robin parameter for general domains. We hope that this new systematic theory will allow broader utilization of optimized Schwarz and 2-Lagrange multiplier preconditioners.

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Section: Motivation For the 2lm Methodmentioning

confidence: 98%

Condition Number Estimates for the Nonoverlapping Optimized Schwarz Method and the 2-Lagrange Multiplier Method for General Domains and Cross Points

Loisel¹

2013

SIAM J. Numer. Anal.

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“…In this section, we summarize previous work done by the author on the analysis and application of the DDM and its optimized variants for solving the elliptic equation on the Yin-Yang grid (Qaddouri, 2008;Qaddouri et al, 2008). Loisel et al (2010) provide an analysis of this optimized DDM for the elliptic equation on the icosahedral grid.…”

Section: Optimized Schwarz Methods For Elliptic Problemmentioning

confidence: 99%

Nonlinear shallow‐water equations on the Yin‐Yang grid

Qaddouri

2011

Quart J Royal Meteoro Soc

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The system of nonlinear shallow-water equations (SWEs) is a hyperbolic system serving as a primary test problem for numerical methods used in modelling global atmospheric flows. In this article, the SWEs on a rotating sphere are solved on the Yin-Yang grid by using a domain decomposition method (DDM). This overset grid is singularity free and has a quasi-uniform grid spacing. It is composed of two identical latitude/longitude orthogonal grid panels that are combined to cover the sphere with partial overlap on their boundaries. On each of the two subgrids, the local solver is based on an implicit and semi-Lagrangian discretization on a horizontally staggered Arakawa C mesh. The resulting positive definite Helmholtz problem is solved using a Schwarz-type DDM known as the optimized Schwarz method, which gives better performance than the classical Schwarz method by using specific Robin or higher-order transmission conditions. Finally, the standard shallow-water test set is performed in order to show that the DDM solution for SWEs on the Yin-Yang grid system can reproduce the global solution accurately on the sphere. This work represents a first step in the development of a three-dimensional forecasting model on the Yin-Yang grid.

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“…Let 1 ≤ p < ∞. Assume g ∈ W 1, p (25), to minimize J (u) over u ∈ V h , starting from (u, s) = (0,ŝ) given by (31), converges in at most N * iterations, where…”

Section: Introductionmentioning

confidence: 99%