Results 1  10
of
185
Balancing Domain Decomposition
 Comm. Numer. Meth. Engrg
, 1993
"... The NeumannNeumann algorithm is known to be an efficient domain decomposition preconditioner with unstructured subdomains for iterative solution of finite element discretizations of difficult problems with strongly discontinuous coefficients [6]. However, this algorithm suffers from the need to sol ..."
Abstract

Cited by 133 (11 self)
 Add to MetaCart
The NeumannNeumann algorithm is known to be an efficient domain decomposition preconditioner with unstructured subdomains for iterative solution of finite element discretizations of difficult problems with strongly discontinuous coefficients [6]. However, this algorithm suffers from the need to solve in each iteration an inconsistent singular problem for every subdomain, and its convergence deteriorates with increasing number of subdomains due to the lack of a coarse problem to propagate the error globally. We show that the equilibrium conditions for the singular problems on subdomains lead to a simple and natural construction of a coarse problem. The construction is purely algebraic and applies also to systems, such as those that arize in elasticity. A convergence bound independent on the number of subdomains is proved and results of computational tests are reported.
Schwarz Analysis Of Iterative Substructuring Algorithms For Elliptic Problems In Three Dimensions
 SIAM J. Numer. Anal
, 1993
"... . Domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations of partial differential equations. The preconditioners are constructed from exact or approximate sol ..."
Abstract

Cited by 110 (26 self)
 Add to MetaCart
. Domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations of partial differential equations. The preconditioners are constructed from exact or approximate solvers for the same partial differential equation restricted to a set of subregions into which the given region has been divided. In addition, the preconditioner is often augmented by a coarse, second level approximation, that provides additional, global exchange of information, and which can enhance the rate of convergence considerably. The iterative substructuring methods, based on decompositions of the region into nonoverlapping subregions, form one of the main families of such algorithms. Many domain decomposition algorithms can conveniently be described and analyzed as Schwarz methods. These algorithms are fully defined in terms of a set of subspaces and auxiliary bilinear forms. A gener...
The Hierarchical Basis Multigrid Method
 Numer. Math
, 1987
"... We derive and analyze the hierarchical basismultigrid method for solving discretizations of selfadjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric GauSeidel iteration with inner iterations, but it is strongly re ..."
Abstract

Cited by 104 (14 self)
 Add to MetaCart
We derive and analyze the hierarchical basismultigrid method for solving discretizations of selfadjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric GauSeidel iteration with inner iterations, but it is strongly related to 2level methods, to the standard multigrid Vcycle, and to earlier Jacobilike hierarchical basis methods. The method is very robust, and has a nearly optimal convergence rate and work estimate. It is especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes. Keywords: hierarchical basis, multigrid, finite elements, adaptive mesh refinement, preconditioned conjugate gradients, symmetric GauSeidel. Subject Classification: AMS(MOS): 65F10, 65F35, 65N20, 65N30 1 1. Introduction In this work we describe and analyse the hierarchical basis multigrid method for solving selfadjoint, positive definite, elliptic bou...
Domain Decomposition Algorithms With Small Overlap
, 1994
"... Numerical experiments have shown that twolevel Schwarz methods often perform very well even if the overlap between neighboring subregions is quite small. This is true to an even greater extent for a related algorithm, due to Barry Smith, where a Schwarz algorithm is applied to the reduced linear ..."
Abstract

Cited by 82 (11 self)
 Add to MetaCart
Numerical experiments have shown that twolevel Schwarz methods often perform very well even if the overlap between neighboring subregions is quite small. This is true to an even greater extent for a related algorithm, due to Barry Smith, where a Schwarz algorithm is applied to the reduced linear system of equations that remains after that the variables interior to the subregions have been eliminated. In this paper, a supporting theory is developed.
Schwarz Methods of NeumannNeumann Type for ThreeDimensional Elliptic Finite Element Problems
 Comm. Pure Appl. Math
, 1995
"... . Several domain decomposition methods of NeumannNeumann type are considered for solving the large linear systems of algebraic equations that arise from discretizations of elliptic problems by finite elements. We will only consider problems in three dimensions. Several new variants of the basic alg ..."
Abstract

Cited by 78 (17 self)
 Add to MetaCart
. Several domain decomposition methods of NeumannNeumann type are considered for solving the large linear systems of algebraic equations that arise from discretizations of elliptic problems by finite elements. We will only consider problems in three dimensions. Several new variants of the basic algorithm are introduced in a Schwarz method framework that provides tools which have already proven very useful in the design and analysis of other domain decomposition and multilevel methods. The NeumannNeumann algorithms have several advantages over other domain decomposition methods. The subregions, which define the subproblems, only share the boundary degrees of freedom with their neighbors. The subregions can also be of quite arbitrary shape and many of the major components of the preconditioner can be constructed from subprograms available in standard finite element program libraries. However, in its original form, the algorithm lacks a mechanism for global transportation of informatio...
Balancing domain decomposition for problems with large jumps in coefficients
 Math. Comp
, 1996
"... Abstract. The Balancing Domain Decomposition algorithm uses in each iteration solution of local problems on the subdomains coupled with a coarse problem that is used to propagate the error globally and to guarantee that the possibly singular local problems are consistent. The abstract theory introdu ..."
Abstract

Cited by 59 (10 self)
 Add to MetaCart
Abstract. The Balancing Domain Decomposition algorithm uses in each iteration solution of local problems on the subdomains coupled with a coarse problem that is used to propagate the error globally and to guarantee that the possibly singular local problems are consistent. The abstract theory introduced recently by the firstnamed author is used to develop condition number bounds for conforming linear elements in two and three dimensions. The bounds are independent of arbitrary coefficient jumps between subdomains and of the number of subdomains, and grow only as the squared logarithm of the mesh size h. Computational experiments for two and threedimensional problems confirm the theory. 1.
DualPrimal Feti Methods For ThreeDimensional Elliptic Problems With Heterogeneous Coefficients
 SIAM J. Numer. Anal
, 2001
"... In this paper, certain iterative substructuring methods with Lagrange multipliers are considered for elliptic problems in three dimensions. The algorithms belong to the family of dualprimal FETI methods which have recently been introduced and analyzed successfully for elliptic problems in the plane ..."
Abstract

Cited by 54 (10 self)
 Add to MetaCart
In this paper, certain iterative substructuring methods with Lagrange multipliers are considered for elliptic problems in three dimensions. The algorithms belong to the family of dualprimal FETI methods which have recently been introduced and analyzed successfully for elliptic problems in the plane. The family of algorithms for three dimensions is extended and a full analysis is provided for the new algorithms. Particular attention is paid to finding algorithms with a small primal subspace since that subspace represents the only global part of the dualprimal preconditioner. It is shown that the condition numbers of several of the dualprimal FETI methods can be bounded polylogarithmically as a function of the dimension of the individual subregion problems and that the bounds are otherwise independent of the number of subdomains, the mesh size, and jumps in the coefficients. These results closely parallel those for other successful iterative substructuring methods of primal as well as dual type.
A comparison of some domain decomposition and ILU preconditioned iterative methods for nonsymmetric elliptic problems
 Numer. Linear Algebra Appl
, 1994
"... In recent years, competitive domaindecomposed preconditioned iterative techniques have beendeveloped for nonsymmetric elliptic problems. In these techniques, a large problem is divided into many smaller problems whose requirements for coordination can be controlled to allow e ective solution on par ..."
Abstract

Cited by 53 (13 self)
 Add to MetaCart
In recent years, competitive domaindecomposed preconditioned iterative techniques have beendeveloped for nonsymmetric elliptic problems. In these techniques, a large problem is divided into many smaller problems whose requirements for coordination can be controlled to allow e ective solution on parallel machines. Acentral question is how tochoose these small problems and how to arrange the order of their solution. Di erent speci cations of decomposition and solution order lead to a plethora of algorithms possessing complementary advantages and disadvantages. In this report we compare several methods, including the additive Schwarz algorithm, the classical multiplicative Schwarz algorithm, an accelerated multiplicative Schwarz algorithm, the tile algorithm, the CGK algorithm, the CSPD algorithm, and also the popular global ILUfamily of preconditioners, on some nonsymmetric or inde nite elliptic model problems discretized by nite di erence methods. The preconditioned problems are solved by the unrestarted GMRES method. A version of the accelerated multiplicative Schwarz method is a consistently good performer.
Overlapping Schwarz Methods On Unstructured Meshes Using NonMatching Coarse Grids
 Numer. Math
, 1996
"... . We consider two level overlapping Schwarz domain decomposition methods for solving the finite element problems that arise from discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Standard finite element interpolation from the coarse to the fine grid may ..."
Abstract

Cited by 49 (17 self)
 Add to MetaCart
. We consider two level overlapping Schwarz domain decomposition methods for solving the finite element problems that arise from discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Standard finite element interpolation from the coarse to the fine grid may be used. Our theory requires no assumption on the substructures which constitute the whole domain, so each substructure can be of arbitrary shape and of different size. The global coarse mesh is allowed to be nonnested to the fine grid on which the discrete problem is to be solved and both the coarse meshes and the fine meshes need not be quasiuniform. In addition, the domains defined by the fine and coarse grid need not be identical. The one important constraint is that the closure of the coarse grid must cover any portion of the fine grid boundary for which Neumann boundary conditions are given. In this general setting, our algorithms have the same optimal convergence rate of the usual ...
Parallelized direct execution simulation of messagepassing parallel programs
 IEEE Transactions on Parallel and Distributed Systems
, 1996
"... As massively parallel computers proliferate, there is growing interest in �nding ways by which performance of massively parallel codes can be e�ciently predicted. This problem arises in diverse contexts such as parallelizing compilers, parallel performance monitoring, and parallel algorithm developm ..."
Abstract

Cited by 45 (10 self)
 Add to MetaCart
As massively parallel computers proliferate, there is growing interest in �nding ways by which performance of massively parallel codes can be e�ciently predicted. This problem arises in diverse contexts such as parallelizing compilers, parallel performance monitoring, and parallel algorithm development. In this paper we describe one solution where one directly executes the application code, but uses a discreteevent simulator to model details of the presumed parallel machine, such as operating system and communication network behavior. Because this approach is computationally expensive, we are interested in its own parallelization, speci�cally the parallelization of the discreteevent simulator. We describe methods suitable for parallelized direct execution simulation of messagepassing parallel programs, and report on the performance of such a system, LAPSE �Large Application Parallel Simulation Environment�, wehave built on the Intel Paragon. On all codes measured to date, LAPSE predicts performance well, typically within 10 � relative error. Depending on the nature of the application code, we have observed low slowdowns �relative to natively executing code � and high relative speedups using up to 64 processors.