Comparison of the Dirichlet-Neumann and Optimal Schwarz Method on the Sphere

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

doi:10.1007/3-540-26825-1_21

Cited by 11 publications

(8 citation statements)

References 9 publications

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“…0.00e + 0 3.79e + 4 0.00e + 0 6.76e + 5 0.00e + 0 2.86e + 6 methods, including RAS, without a coarse grid correction. A quantitative analysis of the asymptotic convergence behavior of the optimized Schwarz methods for different types of differential equations can be found in [4], [6]; see also [13] for a general analysis of optimized Schwarz for the algebraic point of view. We study next the behavior of the convergence of the algebraic optimized Schwarz methods with respect to the change of parameters of the differential equation.…”

Section: Theorem 22 [8]mentioning

confidence: 99%

On the performance of the algebraic optimized Schwarz methods with applications

Laayouni

Szyld

2014

Numer Algor

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Abstract. We investigate the performance of algebraic optimized Schwarz methods used as preconditioners for the solution of discretized differential equations. These methods consist on modifying the so-called transmission blocks. The transmission blocks are replaced by new blocks in order to improve the convergence of the corresponding iterative algorithms. In the optimal case, convergence in two iterations can be achieved. We are also interested in the behavior of the algebraic optimized Schwarz methods with respect to changes in the problems parameters. We focus on constructing preconditioners for different numerically challenging differential problems such as: Periodic and Torus problems; Meshfree problems; Three-dimensional problems. We present different numerical simulations corresponding to different type of problems in two-and three-dimensions.

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Section: Theorem 22 [8]mentioning

confidence: 99%

On the performance of the algebraic optimized Schwarz methods with applications

Laayouni

Szyld

2014

Numer Algor

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“…Written in cylindrical coordinates, the wave equation is 20) where ρ = x 2 + y 2 . Writing (2.20) as…”

Section: Partial Spherical Means Formulas and Boundary Conditions 183mentioning

confidence: 99%

“…Our expression −k(α, β)/ cos β is the m = 1 "fundamental solution" discussed in [20]. It is also closely related to the Green's function for Δ, that is [21] where the prime in (3.14) stands for ∂/∂α differentiation.…”

Section: Associated Poisson Problemmentioning

confidence: 99%

Untitled

2010

Quart. Appl. Math.

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Abstract. We derive various partial spherical means formulas for the 3+1 wave equation. Such formulas, considered earlier by both Weston and Teng, involve only partial integration over a solid angle in addition to history-dependent boundary terms, and are appropriate for faces, edges, and corners of "computational domains". For example, a hemispherical means formula corresponds to a face (plane boundary). Exploiting the theory of wave front sets for linear operators developed by Hörmander, Warchall has proved theorems which suggest the existence of "one-sided update formulas" for wave equations. We attempt to realize such an update formula via an explicit construction based on our hemispherical means formula. We focus on face points and plane boundaries, but also introduce one-fourth and one-eighth spherical means formulas with most of our arguments going through for a point located on either a domain edge or a corner. Throughout our analysis we encounter a number of, we believe, heretofore unknown identities for classical solutions to the wave equation.

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“…It is well-known that overlap usually improves the convergence factor of Schwarz algorithms. Detailed studies for the overlap case do exist for the case of simple domains, such as rectangles, half planes, or hemispheres, and the main analytical tool is the use of Fourier transforms; see, e.g., [3][4][5]8,9,11,18,19,23]. For general domains, and two overlapping subdomains, Kimn [13,14], proved convergence of the method for a model problem with Robin boundary data.…”

Section: Introductionmentioning

confidence: 99%

On the geometric convergence of optimized Schwarz methods with applications to elliptic problems

Loisel

Szyld

2009

Numer. Math.

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The Schwarz method can be used for the iterative solution of elliptic boundary value problems on a large domain . One subdivides into smaller, more manageable, subdomains and solves the differential equation in these subdomains using appropriate boundary conditions. Optimized Schwarz Methods use Robin conditions on the artificial interfaces for information exchange at each iteration, and for which one can optimize the Robin parameters. While the convergence theory of classical Schwarz methods (with Dirichlet conditions on the artificial interface) is well understood, the overlapping Optimized Schwarz Methods still lack a complete theory. In this paper, an abstract Hilbert space version of the Optimized Schwarz Method (OSM) is presented, together with an analysis of conditions for its geometric convergence. It is also shown that if the overlap is relatively uniform, these convergence conditions are met for Optimized Schwarz Methods for two-dimensional elliptic problems, for any positive Robin parameter. In the discrete setting, we obtain that the convergence factor ρ(h) varies like a polylogarithm of h. Numerical experiments show that the methods work well and that the convergence factor does not appear to depend on h. Mathematics Subject Classification (2000)65N55 · 65N22 · 65F10

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Comparison of the Dirichlet-Neumann and Optimal Schwarz Method on the Sphere

Cited by 11 publications

References 9 publications

On the performance of the algebraic optimized Schwarz methods with applications

On the performance of the algebraic optimized Schwarz methods with applications

Untitled

On the geometric convergence of optimized Schwarz methods with applications to elliptic problems

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