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TwoLevel Schwarz Methods for Nonconforming Finite Elements and Discontinuous Coefficients
 Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, Volume 2, number 3224
, 1993
"... . Twolevel domain decomposition methods are developed for a simple nonconforming approximation of second order elliptic problems. A bound is established for the condition number of these iterative methods, which grows only logarithmically with the number of degrees of freedom in each subregion. Thi ..."
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Cited by 23 (1 self)
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. Twolevel domain decomposition methods are developed for a simple nonconforming approximation of second order elliptic problems. A bound is established for the condition number of these iterative methods, which grows only logarithmically with the number of degrees of freedom in each subregion. This bound holds for two and three dimensions and is independent of jumps in the value of the coefficients. Key words. domain decomposition, elliptic problems, preconditioned conjugate gradients, nonconforming finite elements, Schwarz methods AMS(MOS) subject classifications. 65F10, 65N30, 65N55 1. Introduction. The purpose of this paper is to develop a domain decomposition methods for second order elliptic partial differential equations approximated by a simple nonconforming finite element method, the nonconforming P 1 elements. We consider a variant of a twolevel additive Schwarz method introduced in 1987 by Dryja and Widlund [5] for a conforming case. In these methods, a preconditioner i...
A Polylogarithmic Bound For An Iterative Substructuring Method For Spectral Elements In Three Dimensions
 SIAM J. NUMER. ANAL
, 1993
"... Iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems. A pversion finite element method based on continuous, piecewise Q p functions is considered for second order elliptic problems in three dimensions; this special method ..."
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Cited by 21 (4 self)
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Iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems. A pversion finite element method based on continuous, piecewise Q p functions is considered for second order elliptic problems in three dimensions; this special method can also be viewed as a conforming spectral element method. An iterative method is designed for which the condition number of the relevant operator grows only in proportion to (1 + log p)². This bound is independent of jumps in the coefficient of the elliptic problem across the interfaces between the subregions. Numerical results are also reported which support the theory.
Domain decomposition methods for monotone nonlinear elliptic problems
 Contemporary Math
, 1994
"... Abstract. In this paper, we study several overlapping domain decomposition based iterative algorithms for the numerical solution of some nonlinear strongly elliptic equations discretized by the nite element methods. In particular, we consider additive Schwarz algorithms used together with the classi ..."
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Cited by 21 (6 self)
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Abstract. In this paper, we study several overlapping domain decomposition based iterative algorithms for the numerical solution of some nonlinear strongly elliptic equations discretized by the nite element methods. In particular, we consider additive Schwarz algorithms used together with the classical inexact Newton methods. We show that the algorithms converge and the convergence rates are independent of the nite element mesh parameter, as well as the number of subdomains used in the domain decomposition. 1.
Domain Decomposition Algorithms For Mixed Methods For Second Order Elliptic Problems
 Math. Comp
"... . In this paper domain decomposition algorithms for mixed finite element methods for linear and quasilinear second order elliptic problems in IR 2 and IR 3 are developed. A convergence theory for twolevel and multilevel Schwarz methods applied to the algorithms under consideration is given, and ..."
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Cited by 20 (12 self)
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. In this paper domain decomposition algorithms for mixed finite element methods for linear and quasilinear second order elliptic problems in IR 2 and IR 3 are developed. A convergence theory for twolevel and multilevel Schwarz methods applied to the algorithms under consideration is given, and its extension to other substructuring methods such as vertex space and balancing domain decomposition methods is considered. It is shown that the condition number of these iterative methods is bounded uniformly from above in the same manner as in the theory of domain decomposition methods for conforming and nonconforming finite element methods for the same differential problems. Numerical experiments are presented to illustrate the present techniques. 1. Introduction. This is the second paper of a sequence where we develop and analyze efficient iterative algorithms for solving the linear system arising from mixed finite element methods for linear and quasilinear second order elliptic proble...
Different Models Of Parallel Asynchronous Iterations With Overlapping Blocks
 COMPUTATIONAL AND APPLIED MATHEMATICS
, 1998
"... Different computational and mathematical models used in the programming and in the analysis of convergence of asynchronous iterations for parallel solution of linear systems of algebraic equations are studied. The differences between the models are highlighted. Special consideration is given to mode ..."
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Cited by 17 (13 self)
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Different computational and mathematical models used in the programming and in the analysis of convergence of asynchronous iterations for parallel solution of linear systems of algebraic equations are studied. The differences between the models are highlighted. Special consideration is given to models that allow for overlapping blocks, i.e., for the same variable being updated by more than one processor. Keywords. Linear systems, Hmatrices, chaotic relaxation, iterative methods, parallel algorithms, asynchronous algorithms.
Analysis Of Lagrange Multiplier Based Domain Decomposition
, 1998
"... The convergence of a substructuring iterative method with Lagrange multipliers known as Finite Element Tearing and Interconnecting (FETI) method is analyzed in this thesis. This method, originally proposed by Farhat and Roux, decomposes finite element discretization of an elliptic boundary value pro ..."
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Cited by 17 (5 self)
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The convergence of a substructuring iterative method with Lagrange multipliers known as Finite Element Tearing and Interconnecting (FETI) method is analyzed in this thesis. This method, originally proposed by Farhat and Roux, decomposes finite element discretization of an elliptic boundary value problem into Neumann problems on the subdomains, plus a coarse problem for the subdomain null space components. For linear conforming elements and preconditioning by Dirichlet problems on the subdomains, the asymptotic bound on the condition number C(1 log(H=h)) fl , where fl = 2 or 3, is proved for a second order problem, h denoting the characteristic element size and H the size of subdomains. A similar method proposed by Park is shown to be equivalent to FETI with a special choice of some components and the bound C(1 log(H=h)) 2 on the condition number is established. Next, the original FETI method is generalized to fourth order plate bending problems. The main idea there is to enfor...
A minimum overlap restricted additive Schwarz preconditioner and applications in 3D flow simulations
 Contemporary Mathematics
, 1998
"... Numerical simulations of unsteady threedimensional compressible flow problems require the solution of large, sparse, nonlinear systems of equations arising from the discretization of Euler or NavierStokes equations on unstructured, possibly dynamic, meshes. In this ..."
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Cited by 17 (1 self)
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Numerical simulations of unsteady threedimensional compressible flow problems require the solution of large, sparse, nonlinear systems of equations arising from the discretization of Euler or NavierStokes equations on unstructured, possibly dynamic, meshes. In this
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.
QuasiOptimal Schwarz Methods For The Conforming Spectral Element Discretization
, 1995
"... The spectral element method is used to discretize selfadjoint elliptic equations in three dimensional domains. The domain is decomposed into hexahedral elements, and in each of the elements the discretization space is the set of polynomials of degree N in each variable. A conforming Galerkin formul ..."
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Cited by 12 (0 self)
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The spectral element method is used to discretize selfadjoint elliptic equations in three dimensional domains. The domain is decomposed into hexahedral elements, and in each of the elements the discretization space is the set of polynomials of degree N in each variable. A conforming Galerkin formulation is used, the corresponding integrals are computed approximately with GaussLobattoLegendre (GLL) quadrature rules of order N , and a Lagrange interpolation basis associated with the GLL nodes is used. Fast methods are developed for solving the resulting linear system by the preconditioned conjugate gradient method. The conforming finite element space on the GLL mesh, consisting of piecewise Q1 or P1 functions, produces a stiffness matrix Kh that is known to be spectrally equivalent to the spectral element stiffness matrix KN . Kh is replaced by a preconditioner ~ Kh which is well adapted to parallel computer architectures. The preconditioned operator is then ~ K \Gamma1 h KN . ...
Some Recent Results On Schwarz Type Domain Decomposition Algorithms
, 1992
"... Numerical experiments have shown that twolevel Schwarz methods, for the solution of discrete elliptic problems, 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 Sch ..."
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Cited by 10 (1 self)
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Numerical experiments have shown that twolevel Schwarz methods, for the solution of discrete elliptic problems, 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. A supporting theory is outlined.