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27
The Auxiliary Space Method And Optimal Multigrid Preconditioning Techniques For Unstructured Grids
 Computing
, 1996
"... . An abstract framework of auxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxi ..."
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Cited by 31 (2 self)
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. An abstract framework of auxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxiliary space (that may be roughly understood as a nonnested coarser space). An optimal multigrid preconditioner is then obtained for a discretized partial differential operator defined on an unstructured grid by using an auxiliary space defined on a more structured grid in which a further nested multigrid method can be naturally applied. This new technique make it possible to apply multigrid methods to general unstructured grids without too much more programming effort than traditional solution methods. Some simple examples are also given to illustrate the abstract theory and for instance the Morley finite element space is used as an auxiliary space to construct a preconditioner for Argyris ...
A Scalable Substructuring Method By Lagrange Multipliers For Plate Bending Problems
 SIAM J. Numer. Anal
, 1997
"... . We present a new Lagrange multiplier based domain decomposition method for solving iteratively systems of equations arising from the finite element discretization of plate bending problems. The proposed method is essentially an extension of the FETI substructuring algorithm to the biharmonic equat ..."
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Cited by 23 (11 self)
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. We present a new Lagrange multiplier based domain decomposition method for solving iteratively systems of equations arising from the finite element discretization of plate bending problems. The proposed method is essentially an extension of the FETI substructuring algorithm to the biharmonic equation. The main idea is to enforce continuity of the transversal displacement field at the subdomain crosspoints throughout the preconditioned conjugate gradient iterations. The resulting method is proved to have a condition number that does not grow with the number of subdomains, and grows at most polylogarithmically with the number of elements per subdomain. These optimal properties hold for numerous plate bending elements that are used in practice including the HCT, DKT, and a class of nonlocking elements for the ReissnerMindlin plate models. Computational experiments are reported and shown to confirm the theoretical optimal convergence properties of the new domain decomposition method. C...
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...
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...
Intergrid Transfer Operators and Multilevel Preconditioners for Nonconforming Discretizations
, 1997
"... this paper discusses only multigrid (multiplicative) preconditioners and methods). The numbers for the Zienkiewicz element are much more encouraging which makes this element interesting for preconditioning finite element discretization matrices for fourth order problems. ..."
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Cited by 12 (4 self)
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this paper discusses only multigrid (multiplicative) preconditioners and methods). The numbers for the Zienkiewicz element are much more encouraging which makes this element interesting for preconditioning finite element discretization matrices for fourth order problems.
Convergence and optimality of adaptive mixed finite element methods
, 2009
"... The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality. A quasiorthogonality property is proved ..."
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Cited by 8 (3 self)
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The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality. A quasiorthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace, while the part of the error that is not divergence free can be bounded by the data oscillation using a discrete stability result. This discrete stability result is also used to get a localized discrete upper bound which is crucial for the proof of the optimality of the adaptive approximation.
Balancing domain decomposition for nonconforming plate elements
 IMA PREPRINT SERIES, NUMBER 1512
, 1997
"... In this paper the balancing domain decomposition method is extended to nonconforming plate elements. The condition number of the preconditioned system is shown to be bounded by C[1 + ln(H/h)]², where H measures the diameters of the subdomains, h is the mesh size of the triangulation, and the const ..."
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Cited by 8 (3 self)
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In this paper the balancing domain decomposition method is extended to nonconforming plate elements. The condition number of the preconditioned system is shown to be bounded by C[1 + ln(H/h)]², where H measures the diameters of the subdomains, h is the mesh size of the triangulation, and the constant C is independent of H,h and the number of subdomains.
Lower Bounds for TwoLevel Additive Schwarz Preconditioners with Small Overlap
 SIAM J. Sci. Comput
, 1998
"... this paper we will show that for ffi = h (minimal overlap) the following estimate holds for the second order model problem (1:13) (BA h ) c ..."
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Cited by 6 (1 self)
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this paper we will show that for ffi = h (minimal overlap) the following estimate holds for the second order model problem (1:13) (BA h ) c
Schwarz Preconditioners for the Spectral Element Stokes and NavierStokes Discretizations
 Numer. Math
, 1998
"... Introduction We consider fast methods of solving the linear system 8 ! : Au +B t p = f Bu = 0: (1) resulting from the discretization of the Stokes problem by the spectral element method; see (3). The efficient solution of this and analogous systems, generated by a variety of discretization met ..."
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Cited by 4 (0 self)
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Introduction We consider fast methods of solving the linear system 8 ! : Au +B t p = f Bu = 0: (1) resulting from the discretization of the Stokes problem by the spectral element method; see (3). The efficient solution of this and analogous systems, generated by a variety of discretization methods, has been the object of various studies. The Uzawa procedure is a relatively standard technique [GPAR86], and more recently blockdiagonal and blocktriangular preconditioners have been proposed [Elm94, Kla97]. Global pressure variables are used in [BP89] and [TP95] as Lagrange multipliers to constrain the interface velocities and to guarantee that the divergence free condition holds. Rnquist has proposed an iterative substructuring method that is based on a decomposition of the domain into interiors of subregions, faces, edges, and vertices. The coarse problem is a Stoke
The condition number of the Schur complement in domain decomposition
 Numer. Math
, 1999
"... Summary. It is shown that for elliptic boundary value problems of order 2m the condition number of the Schur complement matrix that appears in nonoverlapping domain decomposition methods is of order d 1 h 2m+1, where the parameter d measures the diameters of the subdomains and h is the mesh size of ..."
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Cited by 4 (0 self)
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Summary. It is shown that for elliptic boundary value problems of order 2m the condition number of the Schur complement matrix that appears in nonoverlapping domain decomposition methods is of order d 1 h 2m+1, where the parameter d measures the diameters of the subdomains and h is the mesh size of the triangulation. The result holds for both conforming and nonconforming nite elements. Mathematics Subject Classi cation (1991): 65N55, 65N30. 1.