Multilevel Optimized Schwarz Methods

Gander, Martin J.; Vanzan, Tommaso

doi:10.1137/19m1259389

Cited by 8 publications

(11 citation statements)

References 18 publications

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“…The literature about two-level DD methods is very rich. See, e.g., [5,6,12,20,27,28,29,31,32,33], for references considering DD stationary methods, and, e.g., [1,2,17,19,21,22,23,30,41,40,46,47,50], for references considering DD preconditioners. See also general classical references as [18,45,49] and [36,38].…”

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confidence: 99%

“…On the one hand, many references use different techniques to define coarse functions in the overlap (where the error is predominant), and then extending them on the remaining part of the neighboring subdomains; see, e.g., [17,19,21,22,23,40,41,46,47]. On the other hand, in other works the coarse space is created by first defining basis function on the interfaces (where the residual is non-zero), and then extending them (in different ways) on the portions of the neighboring subdomains; see, e.g., [1,2,5,6,12,27,30,29,31,32,40,33]. For a good, compact and complete overview of several of the different coarse spaces, we refer to [40,Section 5].…”

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Spectral substructured two-level domain decomposition methods

Ciaramella¹,

Vanzan²

2019

Preprint

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Two-level domain decomposition methods are very powerful techniques for the efficient numerical solution of partial differential equations (PDEs). A two-level domain decomposition method requires two main components: a one-level preconditioner (or its corresponding smoothing iterative method), which is based on domain decomposition techniques, and a coarse correction step, which relies on a coarse space. The coarse space must properly represent the error components that the chosen one-level method is not capable to deal with. In the literature most of the works introduced efficient coarse spaces obtained as the span of functions defined on the entire space domain of the considered PDE. Therefore, the corresponding two-level preconditioners and iterative methods are defined in volume.In this paper, a new class of substructured two-level methods is introduced, for which both domain decomposition smoothers and coarse correction steps are defined on the interfaces. This approach has several advantages. On the one hand, the required computational effort is cheaper than the one required by classical volumetric two-level methods. On the other hand, it allows one to use some of the well-known efficient coarse spaces proposed in the literature. Moreover, our new substructured framework can be efficiently extended to a multi-level framework, which is always desirable when the high dimension of the problem or the scarce quality of the coarse space prevents the efficient numerical solution. Numerical experiments demonstrate the effectiveness of the proposed new numerical framework.

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Spectral substructured two-level domain decomposition methods

Ciaramella¹,

Vanzan²

2019

Preprint

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“…Step 2 (fine mesh h = H Finally, we compare the approximate accuracy of the proposed TGDDM1 with recently developed "Two-level Optimized Schwarz method (TLOSM ): Algorithm 2.1 [21]" in Table 5.4 using P 1b − P 1 − P 1 finite elements triple. The TLOSM algorithm execute iteratively several steps on the fine grid to solve the decoupling problems, and then computing the residual equations on the coarse grid to correct the fine grid solutions.…”

Section: Algorithms Hmentioning

confidence: 99%

“…The TLOSM algorithm execute iteratively several steps on the fine grid to solve the decoupling problems, and then computing the residual equations on the coarse grid to correct the fine grid solutions. By following Gander et al [21], we consider the coarse and fine meshes with relationship h = H/2, the fine grid iteration numbers n 1 = n 2 = 2 and the optimized parameters s 1 = 100, s 2 = 1/50. On the other hand, we set δ S = s 1 and δ D = s 2 to simulate the proposed algorithm for the comparison purpose.…”

Section: Algorithms Hmentioning

confidence: 99%

“…Especially, decoupling methods allow us to decompose the coupled problems into several independent subproblems, which can be solved simultaneously with some "legacy" algorithms and codes developed for the Stokes and Darcy equations respectively. We can refer to such decoupling methods as Lagrange multiplier methods [1][2][3], stabilized mixed finite element methods [4][5][6], domain decomposition methods [7][8][9][10][11][12][13][14], two-grid or multi-grid methods [15][16][17][18][19], and optimized Schwarz methods [20,21] and so on.…”

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Two-Grid Domain Decomposition Methods for the Coupled Stokes-Darcy System

Sun,

Shi,

Zheng

et al. 2021

Preprint

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In this paper, we propose two novel Robin-type domain decomposition methods based on the two-grid techniques for the coupled Stokes-Darcy system. Our schemes firstly adopt the existing Robin-type domain decomposition algorithm to obtain the coarse grid approximate solutions. Then two modified domain decomposition methods are further constructed on the fine grid by utilizing the framework of two-grid methods to enhance computational efficiency, via replacing some interface terms by the coarse grid information. The natural idea of using the two-grid frame to optimize the domain decomposition method inherits the best features of both methods and can overcome some of the domain decomposition deficits. The resulting schemes can be implemented easily using many existing mature solvers or codes in a flexible way, which are much effective under smaller mesh sizes or some realistic physical parameters. Moreover, several error estimates are carried out to show the stability and convergence of the schemes. Finally, three numerical experiments are performed and compared with the classical two-grid method, which verifies the validation and efficiency of the proposed algorithms.

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Spectral Coarse Spaces for the Substructured Parallel Schwarz Method

Ciaramella

Vanzan

2022

J Sci Comput

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The parallel Schwarz method (PSM) is an overlapping domain decomposition (DD) method to solve partial differential equations (PDEs). Similarly to classical nonoverlapping DD methods, the PSM admits a substructured formulation, that is, it can be formulated as an iteration acting on variables defined exclusively on the interfaces of the overlapping decomposition. In this manuscript, spectral coarse spaces are considered to improve the convergence and robustness of the substructured PSM. In this framework, the coarse space functions are defined exclusively on the interfaces. This is in contrast to classical two-level volume methods, where the coarse functions are defined in volume, though with local support. The approach presented in this work has several advantages. First, it allows one to use some of the well-known efficient coarse spaces proposed in the literature, and facilitates the numerical construction of efficient coarse spaces. Second, the computational work is comparable or lower than standard volume two-level methods. Third, it opens new interesting perspectives as the analysis of the new two-level substructured method requires the development of a new convergence analysis of general two-level iterative methods. The new analysis casts light on the optimality of coarse spaces: given a fixed dimension m, the spectral coarse space made by the first m dominant eigenvectors is not necessarily the minimizer of the asymptotic convergence factor. Numerical experiments demonstrate the effectiveness of the proposed new numerical framework.

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Multilevel Optimized Schwarz Methods

Cited by 8 publications

References 18 publications

Spectral substructured two-level domain decomposition methods

Spectral substructured two-level domain decomposition methods

Two-Grid Domain Decomposition Methods for the Coupled Stokes-Darcy System

Spectral Coarse Spaces for the Substructured Parallel Schwarz Method

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