In the classical Schwarz framework for conforming approximations of nonsymmetric and indefinite problems [5, 6] the finite element space is optimally decomposed into the sum of a finite number of uniformly overlapped,...
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Optimized Schwarz methods form a class of domain decomposition algorithms in which the transmission conditions are optimized in order to achieve fast convergence. They are usually derived for a model problem with two ...
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ISBN:
(纸本)9783642026775;9783642026768
Optimized Schwarz methods form a class of domain decomposition algorithms in which the transmission conditions are optimized in order to achieve fast convergence. They are usually derived for a model problem with two subdomains, and give efficient transmission conditions for the local coupling between neighboring subdomains. However, when using a large number of subdomains, a coarse space correction is required to achieve parallel scalability. In this paper we demonstrate with a simple model problem that a two-level optimized Schwarz preconditioner is much more effective than a corresponding two-level Restricted Additive Schwarz preconditioner. The weak dependence on the mesh size is retained from the one-level method, while gaining independence on the number of subdomains. Moreover, the best Robin transmission condition is well approximated by using the analysis from the two subdomain case, under Krylov acceleration.
The strategy of domain decomposition methods is to decompose the computational domain into smaller subdomains. Each subdomain is assigned to one processor. The equations are solved on each subdomain. In order to enfor...
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ISBN:
(纸本)9783642026775;9783642026768
The strategy of domain decomposition methods is to decompose the computational domain into smaller subdomains. Each subdomain is assigned to one processor. The equations are solved on each subdomain. In order to enforce the matching of the local solutions, interface conditions have to be written on the boundary between subdomains. These conditions are imposed iteratively. The convergence rate is very sensitive to these interface conditions. The Schwarz method is based on the use of Dirichlet boundary conditions. It can be slow and requires overlapping decompositions. In order to improve the convergence and to be able to use non-overlapping decompositions, it has been proposed to use more general boundary conditions. It is even possible to optimize them with respect to the efficiency of the method. Theoretical and numerical results are given along with open problems.
In this paper, we consider a new approach to estimation from below of the lowest eigenvalues of symmetric positive definite elliptic operators. The approach is based on the overlapping domain decomposition procedure a...
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ISBN:
(纸本)9783642026775;9783642026768
In this paper, we consider a new approach to estimation from below of the lowest eigenvalues of symmetric positive definite elliptic operators. The approach is based on the overlapping domain decomposition procedure and on the replacement of subdomain operators by special low rank perturbed scalar operators. The algorithm is illustrated by applications to model problems with mixed boundary conditions and strongly discontinuous coefficients.
In this paper we recall a new domain decomposition method for the Stokes problem obtained via the Smith factorization. From the theoretical point of view, this domain decomposition method is optimal in the sense that ...
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ISBN:
(纸本)9783642026775;9783642026768
In this paper we recall a new domain decomposition method for the Stokes problem obtained via the Smith factorization. From the theoretical point of view, this domain decomposition method is optimal in the sense that it converges in two iterations for a decomposition into two equal domains. Previous results illustrated the fast convergence of the proposed algorithm in some cases. Our algorithm has shown a more robust behavior than Neumann-Neumann or FETI type methods for particular decompositions;as far as general decompositions are concerned, the performances of the three algorithms are similar. Nevertheless, the computations of the singular values of the interface preconditioned problem have shown that one needs a coarse space whose dimension is less than the one needed for the Neumann-Neumann algorithm. In this work we present a new strategy in order to improve the convergence of the new algorithm in the presence of cross points.
We consider the information transfer between non-matching finite element meshes arising from domain decomposition. Dealing with complex three-dimensional geometries, especially in the case of computational mechanics a...
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ISBN:
(纸本)9783642026775;9783642026768
We consider the information transfer between non-matching finite element meshes arising from domain decomposition. Dealing with complex three-dimensional geometries, especially in the case of computational mechanics and nonlinear contact problems, one can usually not achieve a decomposition of the global domain with mere planar interfaces in a sensible way. Thus, subdomains with warped interfaces emerge which, after an independent discretization, yield a geometrically non-conforming decomposition with small gaps and overlaps. In this paper, we employ a mortar approach and develop a method for the assembly of a discrete coupling operator providing a stable information transfer across such geometrically distinct warped interfaces.
In this paper, we present an additive Neumann-Neumann type parallel method for solving the system of algebraic equations arising from the mortar finite element discretization of a plate problem on a nonconforming mesh...
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ISBN:
(纸本)9783642026775;9783642026768
In this paper, we present an additive Neumann-Neumann type parallel method for solving the system of algebraic equations arising from the mortar finite element discretization of a plate problem on a nonconforming mesh. Locally, we use a conforming Hsieh-Clough-Tocher macro element in the subdomains. The proposed method is almost optimal i.e. the condition number of the preconditioned problem grows poly-logarithmically with respect to the parametes of the local triangulations.
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