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33
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 35 (10 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.
A preconditioner for solving the inner problem of the pversion of the FEM, Part II algebraic multigrid proof
, 2001
"... Finding a fast solver for the inner problem in a DD preconditioner for the pversion of the FEM is a difficult question. We discovered, that the system matrix for the inner problem in 2D has a similar structure to matrices resulting from discretizations of y 2 u xx x 2 u yy in the unit square using ..."
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Cited by 19 (6 self)
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Finding a fast solver for the inner problem in a DD preconditioner for the pversion of the FEM is a difficult question. We discovered, that the system matrix for the inner problem in 2D has a similar structure to matrices resulting from discretizations of y 2 u xx x 2 u yy in the unit square using hversion of the FEM or finite differences. Applying multigrid methods with special smoothers, we have a fast solver for the pversion of the FEM. We give a convergence proof for the multigrid method and the AMLImethod and present some numerical experiments confirming the theory.
Analysis of nonoverlapping domain decomposition algorithms with inexact solves
 Mathematics of Computation
, 1998
"... Abstract. In this paper we construct and analyze new nonoverlapping domain decomposition preconditioners for the solution of secondorder elliptic and parabolic boundary value problems. The preconditioners are developed using uniform preconditioners on the subdomains instead of exact solves. They e ..."
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Cited by 16 (0 self)
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Abstract. In this paper we construct and analyze new nonoverlapping domain decomposition preconditioners for the solution of secondorder elliptic and parabolic boundary value problems. The preconditioners are developed using uniform preconditioners on the subdomains instead of exact solves. They exhibit the same asymptotic condition number growth as the corresponding preconditioners with exact subdomain solves and are much more efficient computationally. Moreover, this asymptotic condition number growth is bounded independently of jumps in the operator coefficients across subdomain boundaries. We also show that our preconditioners fit into the additive Schwarz framework with appropriately chosen subspace decompositions. Condition numbers associated with the new algorithms are computed numerically in several cases and compared with those of the corresponding algorithms in which exact subdomain solves are used. 1.
Comparison of Parallel Solvers for Nonlinear Elliptic Problems Based on Domain Decomposition Ideas
 PARALLEL COMPUTING
, 1995
"... In the present paper, the solution of nonlinear elliptic boundary value problems (b.v.p.) on parallel machines with Multiple Instruction Multiple Data (MIMD) architecture is discussed. Especially, we consider electromagnetic field problems the numerical solution of which is based on finite elem ..."
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Cited by 16 (5 self)
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In the present paper, the solution of nonlinear elliptic boundary value problems (b.v.p.) on parallel machines with Multiple Instruction Multiple Data (MIMD) architecture is discussed. Especially, we consider electromagnetic field problems the numerical solution of which is based on finite element discretizations and a nested Newton solver. For solving the linear systems of algebraic finite element equations in each Newton step, parallel conjugate gradient methods with a Domain Decomposition preconditioner (DD PCG) as well as parallelized global multigrid methods are applied. The implementation of the whole algorithm, i.e. the mesh generation, the generation of the finite element equations, the nested Newton algorithm, the DD PCG method and the global multigrid method, is based on a nonoverlapping DD data structure. The efficiency of the parallel DD PCG methods and the parallelized global multigrid methods, which are embedded in the nested Newton solver, are compared. Fur...
Adaptive Domain Decomposition Methods for Finite and Boundary Element Equations
 In Reports from the Final Conference of the Priority Research Programme Boundary Element Methods 19891995
, 1995
"... The use of the FEM and BEM in different subdomains of a nonoverlapping Domain Decomposition (DD) and their coupling over the coupling boundaries (interfaces) brings about several advantages in many practical applications. The paper presents parallel solvers for largescaled coupled FEBEDD equa ..."
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Cited by 13 (9 self)
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The use of the FEM and BEM in different subdomains of a nonoverlapping Domain Decomposition (DD) and their coupling over the coupling boundaries (interfaces) brings about several advantages in many practical applications. The paper presents parallel solvers for largescaled coupled FEBEDD equations approximating linear and nonlinear plane magnetic field problems as well as plane linear elasticity problems. The parallel algorithms presented are of asymptotically optimal, or, at least, almost optimal complexity and of high parallel efficiency. Key words: Linear and nonlinear elliptic boundary value problems, magnetic field problems, elasticity problems, domain decomposition, finite elements, boundary elements, coupling, solvers, preconditioners, parallel algorithms AMS (MOS) subject classification: 65N55, 65N22, 65F10, 65N30, 65N38, 65Y05, 65Y10 1 Introduction The Domain Decomposition (DD) approach offers many opportunities to marry the advantages of the Finite Element Method (F...
Hierarchical Extension Operators plus Smoothing in Domain Decomposition Preconditioners
 Applied Numerical Mathematics
, 1997
"... The paper presents a cheap technique for the approximation of the harmonic extension from the boundary into the interior of a domain with respect to a given differential operator. The new extension operator is based on the hierarchical splitting of the given f.e. space together with smoothing sweeps ..."
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Cited by 8 (6 self)
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The paper presents a cheap technique for the approximation of the harmonic extension from the boundary into the interior of a domain with respect to a given differential operator. The new extension operator is based on the hierarchical splitting of the given f.e. space together with smoothing sweeps and an exact discrete harmonic extension on the lowest level and will be used as a component in a domain decomposition (DD) preconditioner. In combination with an additional algorithmical improvement of this DDpreconditioner solution times faster then the previously studied were achieved for the preconditioned parallelized cgmethod. The analysis of the new extension operator gives the result that in the 2Dcase O(ln(ln(h \Gamma1 ))) smoothing sweeps per level are sufficient to achieve an hindependent behavior of the preconditioned system provided that there exists a spectrally equivalent preconditioner for the modified Schur complement with spectral equivalence constants independent o...
Parallel Implementation of a Multiblock Method with Approximate Subdomain Solution
, 1998
"... Solution of large linear systems encountered in computational fluid dynamics often naturally leads to some form of domain decomposition, especially when it is desired to use parallel machines. It has been proposed to use approximate solvers to obtain fast but rough solutions on the separate subdomai ..."
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Cited by 8 (4 self)
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Solution of large linear systems encountered in computational fluid dynamics often naturally leads to some form of domain decomposition, especially when it is desired to use parallel machines. It has been proposed to use approximate solvers to obtain fast but rough solutions on the separate subdomains. In this paper a number of approximate solvers are considered, and numerical experiments are included showing speedups obtained on a cluster of workstations as well as on a distributed memory parallel computer. Additionally, some remarks are made pertaining to the practical application of Householder reflections as an orthogonalization procedure within Krylov subspace methods.
A Modular Algebraic Multilevel Method
 IN VIRTUAL PROCEEDINGS OF THE NINTH INTERNATIONAL GAMMWORKSHOP ON PARALLEL MULTIGRID METHODS
, 1996
"... In this paper, we propose a modular algebraic multilevel method on unstructured meshes which is intended to generalize known methods on rectangular meshes. To define the transfer operators, we utilize the block structure of the matrix which is induced by the finecoarse partitioning of the matrix gr ..."
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Cited by 8 (1 self)
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In this paper, we propose a modular algebraic multilevel method on unstructured meshes which is intended to generalize known methods on rectangular meshes. To define the transfer operators, we utilize the block structure of the matrix which is induced by the finecoarse partitioning of the matrix graph and approximate the block of finefine couplings by a modification of its lower, respectively upper triangular part. Numerical experiments show that this approach yields a working preconditioner. Its efficiency in the current implementation depends on the amount of structure information given on input.
Parallel Solution of Finite Element Equation Systems: Efficient InterProcessor Communication
, 1995
"... . This paper deals with the application of domain decomposition methods for the parallel solution of boundary value problems for partial differential equations over a domain\Omega ae IR d , d = 2; 3. The attention is focused on the conception of efficient communication routines for the data excha ..."
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Cited by 7 (4 self)
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. This paper deals with the application of domain decomposition methods for the parallel solution of boundary value problems for partial differential equations over a domain\Omega ae IR d , d = 2; 3. The attention is focused on the conception of efficient communication routines for the data exchange which is necessary for example in the preconditioned cgalgorithm for solving the resulting system of algebraic equations. The paper describes the data structure, different algorithms, and computational tests. Key Words. Parallel computation, finite element method, domain decomposition. AMS(MOS) subject classification. 65Y05, 65Y10, 68M07 Contents 1 Introduction 2 2 Notation and data representation 2 3 Accumulation of data at crosspoints 3 4 Accumulation of data belonging to edges (2D) and faces (3D) 4 5 Accumulation of data belonging to edges (3D) 5 6 Unification of the accumulation of Kettes in two and three dimensions 8 7 Numerical comparison of the algorithms 9 References 9 The ...
Parallelization of MultiGrid Methods Based on Domain Decomposition Ideas
, 1995
"... In the paper, the parallelization of multigrid methods for solving secondorder elliptic boundary value problems in twodimensional domains is discussed. The parallelization strategy is based on a nonoverlapping domain decomposition data structure such that the algorithm is wellsuited for an impl ..."
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Cited by 7 (1 self)
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In the paper, the parallelization of multigrid methods for solving secondorder elliptic boundary value problems in twodimensional domains is discussed. The parallelization strategy is based on a nonoverlapping domain decomposition data structure such that the algorithm is wellsuited for an implementation on a parallel machine with MIMD architecture. For getting an algorithm with a good parallel performance it is necessary to have as few communication as possible between the processors. In our implementation, communication is only needed within the smoothing procedures and the coarsegrid solver. The interpolation and restriction procedures can be performed without any communication. New variants of smoothers of GaussSeidel type having the same communication cost as Jacobi smoothers are presented. For solving the coarsegrid systems iterative methods are proposed that are applied to the corresponding Schur complement system. Three numerical examples, namely a Poisson equation, a m...