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A nonoverlapping domain decomposition method for Maxwell’s equations in three dimensions
- SIAM J. Numer. Anal
"... Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with muc ..."
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Cited by 48 (16 self)
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Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with much simpler coarse solvers. Though the condition number of the preconditioned system may not have a good bound, we are able to show that the convergence rate of the PCG method with such substructuring preconditioner is nearly optimal, and also robust with respect to the (possibly large) jumps of the coefficient in the elliptic equation. 1.
Inexact data-sparse boundary element tearing and interconnecting methods
- SIAM JOURNAL ON SCIENTIFIC COMPUTING
, 2007
"... The Boundary Element Tearing and Interconnecting (BETI) methods have recently been introduced as boundary element counterparts of the well–established Finite Element Tearing and Interconnecting (FETI) methods. In this paper we present inexact data–sparse versions of the BETI methods which avoid the ..."
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Cited by 7 (6 self)
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The Boundary Element Tearing and Interconnecting (BETI) methods have recently been introduced as boundary element counterparts of the well–established Finite Element Tearing and Interconnecting (FETI) methods. In this paper we present inexact data–sparse versions of the BETI methods which avoid the elimination of the primal unknowns and dense matrices. However, instead of symmetric and positive definite systems, we finally have to solve two–fold saddle point problems. The proposed iterative solvers and preconditioners result in almost optimal solvers the complexity of which is proportional to the number of unknowns on the skeleton up to some polylogarithmical factor. Moreover, the solvers are robust with respect to large coefficient jumps.
A domain decomposition solver for a parallel adaptive meshing paradigm
- SIAM J. SCI. COMPUT
"... We describe a domain decomposition algorithm for use in the parallel adaptive meshing paradigm of Bank and Holst. Our algorithm has low communication, makes extensive use of existing sequential solvers, and exploits in several important ways data generated as part of the adaptive meshing paradigm. ..."
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Cited by 6 (1 self)
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We describe a domain decomposition algorithm for use in the parallel adaptive meshing paradigm of Bank and Holst. Our algorithm has low communication, makes extensive use of existing sequential solvers, and exploits in several important ways data generated as part of the adaptive meshing paradigm. Numerical examples illustrate the effectiveness of the procedure.
Extension operators on tensor product structures
- in 2d and 3d. Applied Numerical Mathematics, (available online
, 2004
"... In this paper, a uniformly elliptic second order boundary value problem in 2D is discretized by the p-version of the finite element method. An inexact Dirichlet-Dirichlet domain decomposition pre-conditioner for the system of linear algebraic equations is investigated. The ingredients of such a pre- ..."
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Cited by 6 (5 self)
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In this paper, a uniformly elliptic second order boundary value problem in 2D is discretized by the p-version of the finite element method. An inexact Dirichlet-Dirichlet domain decomposition pre-conditioner for the system of linear algebraic equations is investigated. The ingredients of such a pre-conditioner are an pre-conditioner for the Schur complement, an preconditioner for the sub-domains and an extension operator operating from the edges of the elements into their interior. Using methods of multi-resolution analysis, we propose a new method in order to compute the extension efficiently. We prove that this type of extension is optimal, i.e. the H 1 (Ω)-norm of the extended function is bounded by the H 0.5 (∂Ω)-norm of the given function. Numerical experiments show the optimal performance of the described extension. 1
Extension Theorems For Stokes And Lamé Equations For Nearly Incompressible Media And Their Applications To Numerical Solution Of Problems With Highly Discontinuous Coefficients
"... . We prove extension theorems in the norms described by Stokes and Lame operators for the threedimensional case with periodic boundary conditions. For the Lame equations, we show that the extension theorem holds for nearly incompressible media, but may fail in the opposite limit, i.e., for case of a ..."
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Cited by 3 (3 self)
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. We prove extension theorems in the norms described by Stokes and Lame operators for the threedimensional case with periodic boundary conditions. For the Lame equations, we show that the extension theorem holds for nearly incompressible media, but may fail in the opposite limit, i.e., for case of absolutely compressible media. We study carefully the latter case and associate it with the Cosserat problem. Extension theorems serve as an important tool in many applications, e.g., in domain decomposition and ctitious domain methods, and in analysis of the nite element methods. We consider an application of established extension theorems to an ecient iterative solution technique for the isotropic linear elasticity equations for nearly incompressible media and for the Stokes equations with highly discontinuous coecients. The iterative method involves a special choice for an initial guess and a preconditioner based on solving a constant coecient problem. Such preconditioner allows the us...
A Weakly Overlapping Domain Decomposition for the Adaptive Finite Element Solution of Elliptic Partial Differential Equations
- SIAM J. Numer. Anal
, 1999
"... We present a new domain decomposition preconditioner of additive Schwartz type which is appropriate for use in the parallel adaptive finite element solution of elliptic partial differential equations (PDEs). As with most parallel domain decomposition methods each processor may be assigned one or mor ..."
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Cited by 3 (3 self)
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We present a new domain decomposition preconditioner of additive Schwartz type which is appropriate for use in the parallel adaptive finite element solution of elliptic partial differential equations (PDEs). As with most parallel domain decomposition methods each processor may be assigned one or more subdomains and the preconditioner is such that the processors are able to solve their own subproblem(s) concurrently. The novel feature of the technique proposed here is that it requires just a single layer of overlap in the elements which make up each subdomain at each level of refinement, and it is shown that this amount of overlap is sucient to yield an optimal preconditioner. Some numerical experiments are included to confirm that the condition number when using the new preconditioner is indeed independent of the level of mesh refinement on the test problems considered, and also that it is significantly more computationally efficient than a more conventional additive Schwartz preconditioner which uses an overlap width which is independent of the level of refinement in the mesh (which is well-known to be optimal).
Applied Numerical Mathematics 43 (2002) 211–227 Construction of explicit extension operators
"... on general finite element grids ..."
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Domain Decomposition Preconditioning for Elliptic Problems with Jumps in Coefficients
"... In this paper, we propose an effective iterative preconditioning method to solve elliptic problems with jumps in coefficients. The algorithm is based on the additive Schwarz method (ASM). First, we consider a domain decomposition method without ‘cross points’ on interfaces between subdomains and the ..."
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In this paper, we propose an effective iterative preconditioning method to solve elliptic problems with jumps in coefficients. The algorithm is based on the additive Schwarz method (ASM). First, we consider a domain decomposition method without ‘cross points’ on interfaces between subdomains and the second is the ‘cross points ’ case. In both cases the main computational cost is an implementation of preconditioners for the Laplace operator in whole domain and in subdomains. Iterative convergence is independent of jumps in coefficients and mesh size. 1
Domain Decomposition Preconditioning for Elliptic Problems with Jumps in Coefficients Powered by TCPDF (www.tcpdf.org) Domain Decomposition Preconditioning for Elliptic Problems with Jumps in Coefficients
"... In this paper, we propose an effective iterative preconditioning method to solve elliptic problems with jumps in coefficients. The algorithm is based on the additive Schwarz method (ASM). First, we consider a domain decomposition method without ‘cross points’ on interfaces between subdomains and the ..."
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In this paper, we propose an effective iterative preconditioning method to solve elliptic problems with jumps in coefficients. The algorithm is based on the additive Schwarz method (ASM). First, we consider a domain decomposition method without ‘cross points’ on interfaces between subdomains and the second is the ‘cross points ’ case. In both cases the main computational cost is an implementation of preconditioners for the Laplace operator in whole domain and in subdomains. Iterative convergence is independent of jumps in coefficients and mesh size. 1
