On Monotone and Geometric Convergence of Schwarz Methods for Two-Sided Obstacle Problems

Zeng, Jinshan; Zhou, Shuai

doi:10.1137/s0036142995288920

Cited by 45 publications

(38 citation statements)

References 7 publications

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“…for some D > 0 independent of h, δ and m. Then, the convergence factor given by (38) 2 }, we have < 1, independent of h and m. Finally, we note that the convergence factor for some synchronous overlapping domain decomposition without the coarse mesh has been studied in [1,60]. The schemes obtained from our algorithms are different from those of [1,60] in the treatment of the subproblem obstacles.…”

Section: Applications To Obstacle Problemsmentioning

confidence: 90%

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Convergence rate analysis of an asynchronous space decomposition method for convex Minimization

Tai

Tseng

2001

Math. Comp.

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Abstract. We analyze the convergence rate of an asynchronous space decomposition method for constrained convex minimization in a reflexive Banach space. This method includes as special cases parallel domain decomposition methods and multigrid methods for solving elliptic partial differential equations. In particular, the method generalizes the additive Schwarz domain decomposition methods to allow for asynchronous updates. It also generalizes the BPX multigrid method to allow for use as solvers instead of as preconditioners, possibly with asynchronous updates, and is applicable to nonlinear problems. Applications to an overlapping domain decomposition for obstacle problems are also studied. The method of this work is also closely related to relaxation methods for nonlinear network flow. Accordingly, we specialize our convergence rate results to the above methods. The asynchronous method is implementable in a multiprocessor system, allowing for communication and computation delays among the processors.

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Section: Applications To Obstacle Problemsmentioning

confidence: 90%

“…The schemes obtained from our algorithms are different from those of [1,60] in the treatment of the subproblem obstacles. The algorithms of [1,60] use the global obstacle for the subdomain problems. In our algorithms, the subdomain obstacles can be updated dynamically during the iterations.…”

Section: Applications To Obstacle Problemsmentioning

confidence: 96%

Convergence rate analysis of an asynchronous space decomposition method for convex Minimization

Tai

Tseng

2001

Math. Comp.

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“…For convergence of discrete Schwarz algorithms of either additive or multiplicative types, see for example, [1,6,7,11].…”

Section: Introductionmentioning

confidence: 99%

Maximum norm analysis of an overlapping nonmatching grids method for the obstacle problem

Boulbrachene

Saadi

2006

Adv. Differ Equ.

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We provide a maximum norm analysis of an overlapping Schwarz method on nonmatching grids for second-order elliptic obstacle problem. We consider a domain which is the union of two overlapping subdomains where each subdomain has its own independently generated grid. The grid points on the subdomain boundaries need not match the grid points from the other subdomain. Under a discrete maximum principle, we show that the discretization on each subdomain converges quasi-optimally in the L ∞ norm.

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“…For problems related to variational inequalities, there are results by Lions [14], Hoffman and Zou [8], Kuznetsov and Neittaanmäki [10], Kuznetsov, Neittaanmäki and Tarvainen [11]- [12], Lü, Liem and Shih [18], Tarvainen [19], Badea [2], and Zeng and Zhou [20] in the domain decomposition method, and by Kornhuber [9] and Mandel [13] in the multigrid method. Despite those results, the functioning and convergence of the Schwarz domain decomposition method is not fully understood for variational inequalities.…”

Section: Introductionmentioning

confidence: 99%

An additive Schwarz method for variational inequalities

Badea

Wang

1999

Math. Comp.

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Abstract. This paper proposes an additive Schwarz method for variational inequalities and their approximations by finite element methods. The Schwarz domain decomposition method is proved to converge with a geometric rate depending on the decomposition of the domain. The result is based on an abstract framework of convergence analysis established for general variational inequalities in Hilbert spaces.

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On Monotone and Geometric Convergence of Schwarz Methods for Two-Sided Obstacle Problems

Cited by 45 publications

References 7 publications

Convergence rate analysis of an asynchronous space decomposition method for convex Minimization

Convergence rate analysis of an asynchronous space decomposition method for convex Minimization

Maximum norm analysis of an overlapping nonmatching grids method for the obstacle problem

An additive Schwarz method for variational inequalities

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