Results 1  10
of
98
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 ..."
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Cited by 82 (11 self)
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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 ..."
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Cited by 78 (17 self)
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. 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...
Feti And NeumannNeumann Iterative Substructuring Methods: Connections And New Results
 Comm. Pure Appl. Math
, 1999
"... The FETI and NeumannNeumann families of algorithms are among the best known and most severely tested domain decomposition methods for elliptic partial differential equations. They are iterative substructuring methods and have many algorithmic components in common but there are also differences. The ..."
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Cited by 60 (17 self)
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The FETI and NeumannNeumann families of algorithms are among the best known and most severely tested domain decomposition methods for elliptic partial differential equations. They are iterative substructuring methods and have many algorithmic components in common but there are also differences. The purpose of this paper is to further unify the theory for these two families of methods and to introduce a new family of FETI algorithms. Bounds on the rate of convergence, which are uniform with respect to the coefficients of a family of elliptic problems with heterogeneous coefficients, are established for these new algorithms. The theory for a variant of the NeumannNeumann algorithm is also redeveloped stressing similarities to that for the FETI methods.
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 ..."
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Cited by 54 (13 self)
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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.
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 ..."
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Cited by 53 (10 self)
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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.
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 ..."
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Cited by 49 (17 self)
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. 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 ...
Tensor Product Type Subspace Splittings And Multilevel Iterative Methods For Anisotropic Problems
, 1994
"... We describe tensor product type techniques to derive robust solvers for anisotropic elliptic model problems on rectangular domains in R d . Our analysis is based on the theory of additive subspace correction methods and applies to finiteelement and prewaveletschemes. We present multilevel and ..."
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Cited by 45 (17 self)
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We describe tensor product type techniques to derive robust solvers for anisotropic elliptic model problems on rectangular domains in R d . Our analysis is based on the theory of additive subspace correction methods and applies to finiteelement and prewaveletschemes. We present multilevel and prewaveletbased methods that are robust for anisotropic diffusion operators with additional Helmholtz term. Furthermore the resulting convergence rates are independent of the discretization level. Beside their theoretical foundation, we also report on the results of various numerical experiments to compare the different methods.
Parallel NewtonKrylovSchwarz Algorithms For The Transonic Full Potential Equation
, 1998
"... We study parallel twolevel overlapping Schwarz algorithms for solving nonlinear finite element problems, in particular, for the full potential equation of aerodynamics discretized in two dimensions with bilinear elements. The overall algorithm, NewtonKrylovSchwarz (NKS), employs an inexact finite ..."
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Cited by 42 (27 self)
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We study parallel twolevel overlapping Schwarz algorithms for solving nonlinear finite element problems, in particular, for the full potential equation of aerodynamics discretized in two dimensions with bilinear elements. The overall algorithm, NewtonKrylovSchwarz (NKS), employs an inexact finitedifference Newton method and a Krylov space iterative method, with a twolevel overlapping Schwarz method as a preconditioner. We demonstrate that NKS, combined with a density upwinding continuation strategy for problems with weak shocks, is robust and economical for this class of mixed elliptichyperbolic nonlinear partial differential equations, with proper specification of several parameters. We study upwinding parameters, inner convergence tolerance, coarse grid density, subdomain overlap, and the level of fillin in the incomplete factorization, and report their effect on numerical convergence rate, overall execution time, and parallel efficiency on a distributedmemory parallel computer.
An Optimal Preconditioner for a Class of Saddle Point Problems with a Penalty Term
 SIAM J. Sci. Comput
, 1995
"... . Iterative methods are considered for a class of saddle point problems with a penalty term arising from finite element discretizations of certain elliptic problems. An optimal preconditioner which is independent of the discretization and the penalty parameter is constructed. This approach is then u ..."
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Cited by 40 (4 self)
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. Iterative methods are considered for a class of saddle point problems with a penalty term arising from finite element discretizations of certain elliptic problems. An optimal preconditioner which is independent of the discretization and the penalty parameter is constructed. This approach is then used to design an iterative method with a convergence rate independent of the Lam'e parameters occuring in the equations of linear elasticity. Key words. mixed finite elements, saddle point problems, penalty term, nearly incompressible materials, elasticity, preconditioned conjugate residual method, domain decomposition, multigrid AMS(MOS) subject classifications. 65F10, 65N22, 65N30, 65N55, 73V05 1. Introduction. In recent years, modern iterative methods, e.g. domain decomposition and multigrid methods, have been applied to parameter dependent problems arising in solid mechanics; see Braess [6], Braess and Blomer [8], Jung [28], and Smith [38]. If the direct approach of (low order) conform...
An EnergyMinimizing Interpolation For Robust Multigrid Methods
 SIAM J. SCI. COMPUT
, 1998
"... We propose a robust interpolation for multigrid based on the concepts of energy minimization and approximation. The formulation is general; it can be applied to any dimensions. The analysis for one dimension proves that the convergence rate of the resulting multigrid method is independent of the coe ..."
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Cited by 37 (6 self)
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We propose a robust interpolation for multigrid based on the concepts of energy minimization and approximation. The formulation is general; it can be applied to any dimensions. The analysis for one dimension proves that the convergence rate of the resulting multigrid method is independent of the coefficient of the underlying PDE, in addition to being independent of the mesh size. We demonstrate numerically the effectiveness of the multigrid method in two dimensions by applying it to a discontinuous coefficient problem and an oscillatory coefficient problem. We also show using a onedimensional Helmholtz problem that the energy minimization principle can be applied to solving elliptic problems that are not positive definite.