Results 1  10
of
81
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...
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...
Hierarchical Bases and the Finite Element Method
, 1997
"... CONTENTS 1 Introduction 1 2 Preliminaries 3 3 Fundamental TwoLevel Estimates 7 4 A Posteriori Error Estimates 16 5 TwoLevel Iterative Methods 23 6 Multilevel Cauchy Inequalities 30 7 Multilevel Iterative Methods 34 References 41 1. Introduction In this work we present a brief introduction to hie ..."
Abstract

Cited by 61 (3 self)
 Add to MetaCart
CONTENTS 1 Introduction 1 2 Preliminaries 3 3 Fundamental TwoLevel Estimates 7 4 A Posteriori Error Estimates 16 5 TwoLevel Iterative Methods 23 6 Multilevel Cauchy Inequalities 30 7 Multilevel Iterative Methods 34 References 41 1. Introduction In this work we present a brief introduction to hierarchical bases, and the important part they play in contemporary finite element calculations. In particular, we examine their role in a posteriori error estimation, and in the Department of Mathematics, University of California at San Diego, La Jolla, CA 92093. The work of this author was supported by the Office of Naval Research under contract N0001489J1440. 2 Randolph E. Bank formulation of iterative methods for solving the large sparse sets of linear equations arising from the finite element discretization. Our goal is that the development should be largely selfcontained, but at the same time accessible and interest
Multilevel Schwarz Methods For Elliptic Problems With Discontinuous Coefficients In Three Dimensions
 NUMER. MATH
, 1994
"... Multilevel Schwarz methods are developed for a conforming finite element approximation of second order elliptic problems. We focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions. We establish a condition number estimate fo ..."
Abstract

Cited by 54 (13 self)
 Add to MetaCart
Multilevel Schwarz methods are developed for a conforming finite element approximation of second order elliptic problems. We focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions. We establish a condition number estimate for the iterative operator, which is independent of the coefficients, and grows at most as the square of the number of levels. We also characterize a class of distributions of the coefficients, called quasimonotone, for which the weighted L²projection is stable and for which we can use the standard piecewise linear functions as a coarse space. In this case, we obtain optimal methods, i.e. bounds which are independent of the number of levels and subregions. We also design and analyze multilevel methods with new coarse spaces given by simple explicit formulas. We consider nonuniform meshes and conclude by an analysis of multilevel iterative substructuring methods.
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.
New Estimates for Multilevel Algorithms Including the VCycle
 MATH. COMP
, 1993
"... The purpose of this paper is to provide new estimates for certain multilevel algorithms. In particular, we are concerned with the simple additive multilevel algorithm given in [10] and the standard Vcycle algorithm with one smoothing step per grid. We shall prove that these algorithms have a unifo ..."
Abstract

Cited by 49 (5 self)
 Add to MetaCart
The purpose of this paper is to provide new estimates for certain multilevel algorithms. In particular, we are concerned with the simple additive multilevel algorithm given in [10] and the standard Vcycle algorithm with one smoothing step per grid. We shall prove that these algorithms have a uniform reduction per iteration independent of the mesh sizes and number of levels even on nonconvex domains which do not provide full elliptic regularity. For example, the theory applies to the standard multigrid Vcycle on the Lshaped domain or a domain with a crack and yields a uniform convergence rate. We also prove uniform convergence rates for the multigrid Vcycle for problems with nonuniformly refined meshes. Finally, we give a new multigrid approach for problems on domains with curved boundaries and prove a uniform rate of convergence for the corresponding multigrid Vcycle algorithms.
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 ...
WellSpaced Points for Numerical Methods
, 1997
"... mesh generation, mesh coarsening, multigrid Abstract A numerical method for the solution of a partial differential equation (PDE) requires the following steps: (1) discretizing the domain (mesh generation); (2) using an approximation method and the mesh to transform the problem into a linear system; ..."
Abstract

Cited by 44 (2 self)
 Add to MetaCart
mesh generation, mesh coarsening, multigrid Abstract A numerical method for the solution of a partial differential equation (PDE) requires the following steps: (1) discretizing the domain (mesh generation); (2) using an approximation method and the mesh to transform the problem into a linear system; (3) solving the linear system. The approximation error and convergence of the numerical method depend on the geometric quality of the mesh, which in turn depends on the size and shape of its elements. For example, the shape quality of a triangular mesh is measured by its element's aspect ratio. In this work, we shift the focus to the geometric properties of the nodes, rather than the elements, of well shaped meshes. We introduce the concept of wellspaced points and their spacing functions, and show that these enable the development of simple and efficient algorithms for the different stages of the numerical solution of PDEs. We first apply wellspaced point sets and their accompanying technology to mesh coarsening, a crucial step in the multigrid solution of a PDE. A good aspectratio coarsening sequence of an unstructured mesh M0 is a sequence of good aspectratio meshes M1; : : : ; Mk such that Mi is an approximation of Mi\Gamma 1 containing fewer nodes and elements. We present a new approach to coarsening that guarantees the sequence is also of optimal size and width up to a constant factor the first coarsening method that provides these guarantees. We also present experimental results, based on an implementation of our approach, that substantiate the theoretical claims.
A subspace preconditioning algorithm for eigenvector/eigenvalue computation
 Adv. Comput. Math
, 1996
"... We consider the problem of computing a modest number of the smallest eigenvalues along with orthogonal bases for the corresponding eigenspaces of a symmetric positive definite operator A defined on a finite dimensional real Hilbert space V. In our applications, the dimension of V is large and the co ..."
Abstract

Cited by 39 (6 self)
 Add to MetaCart
We consider the problem of computing a modest number of the smallest eigenvalues along with orthogonal bases for the corresponding eigenspaces of a symmetric positive definite operator A defined on a finite dimensional real Hilbert space V. In our applications, the dimension of V is large and the cost of inverting A is prohibitive. In this paper, we shall develop an effective parallelizable technique for computing these eigenvalues and eigenvectors utilizing subspace iteration and preconditioning for A. Estimates will be provided which show that the preconditioned method converges linearly when used with a uniform preconditioner under the assumption that the approximating subspace is close enough to the span of desired eigenvectors. 1. Introduction. In this paper, we shall be concerned with computing a modest number of the smallest eigenvalues and their corresponding eigenvectors of a large symmetric illconditioned system. More explicitly, let A be a symmetric and positive definite linear operator on a Ndimensional real vector space V with inner product (·, ·)