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85
Convergence of Algebraic Multigrid Based on Smoothed Aggregation
 Computing
, 1998
"... . We prove a convergence estimate for the Algebraic Multigrid Method with prolongations defined by aggregation using zero energy modes, followed by a smoothing. The method input is the problem matrix and a matrix of the zero energy modes. The estimate depends only polylogarithmically on the mesh siz ..."
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Cited by 90 (11 self)
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. We prove a convergence estimate for the Algebraic Multigrid Method with prolongations defined by aggregation using zero energy modes, followed by a smoothing. The method input is the problem matrix and a matrix of the zero energy modes. The estimate depends only polylogarithmically on the mesh size, and requires only a weak approximation property for the aggregates, which can be apriori verified computationally. Construction of the prolongator in the case of a general second order system is described, and the assumptions of the theorem are verified for a scalar problem discretized by linear conforming finite elements. Key words. Algebraic multigrid, zero energy modes, convergence theory, computational mechanics, Finite Elements, iterative solvers 1. Introduction. This paper is concerned with the analysis of an Algebraic Multigrid Method (AMG) based on smoothed aggregation, which we have introduced in [28], and which in turn is a further development of [25, 26]. This method and its ...
Algebraic Multigrid Based On Element Interpolation (AMGe)
, 1998
"... We introduce AMGe, an algebraic multigrid method for solving the discrete equations that arise in Ritztype finite element methods for partial differential equations. Assuming access to the element stiffness matrices, AMGe is based on the use of two local measures, which are derived from global meas ..."
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Cited by 81 (13 self)
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We introduce AMGe, an algebraic multigrid method for solving the discrete equations that arise in Ritztype finite element methods for partial differential equations. Assuming access to the element stiffness matrices, AMGe is based on the use of two local measures, which are derived from global measures that appear in existing multigrid theory. These new measures are used to determine local representations of algebraically "smooth" error components that provide the basis for constructing effective interpolation and, hence, the coarsening process for AMG. Here, we focus on the interpolation process; choice of the coarse "grids" based on these measures is the subject of current research. We develop a theoretical foundation for AMGe and present numerical results that demonstrate the efficacy of the method.
FirstOrder System Least Squares For SecondOrder Partial Differential Equations: Part I
, 1994
"... . This paper develops ellipticity estimates and discretization error bounds for elliptic equations (with lower order terms) that are reformulated as a leastsquares problem for an equivalent firstorder system. The main result is the proof of ellipticity, which is used in a companion paper to esta ..."
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Cited by 61 (14 self)
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. This paper develops ellipticity estimates and discretization error bounds for elliptic equations (with lower order terms) that are reformulated as a leastsquares problem for an equivalent firstorder system. The main result is the proof of ellipticity, which is used in a companion paper to establish optimal convergence of multiplicative and additive solvers of the discrete systems. Key words. leastsquares discretization, secondorder elliptic problems, RayleighRitz, finite elements 1. Introduction. The purpose of this paper is to analyse the leastsquares finite element method for secondorder convectiondiffusion equations written as a firstorder system. In general, the standard Galerkin finite element methods applied to nonselfadjoint elliptic equations with significant convection terms exhibit a variety of deficiencies, including oscillations or nonmonotonocity of the solution and poor approximation of its derivatives. A variety of stabilization techniques, such as upwin...
Tetrahedral Grid Refinement
, 1995
"... Zusammenfassung Tetrahedral Grid Refinement. We present a refinement algorithm for unstructured tetrahedral grids, which generates possibly highly nonuniform but nevertheless consistent (closed) and stable triangulations. Therefore we first define some local regular and irregular refinement rules t ..."
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Cited by 51 (1 self)
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Zusammenfassung Tetrahedral Grid Refinement. We present a refinement algorithm for unstructured tetrahedral grids, which generates possibly highly nonuniform but nevertheless consistent (closed) and stable triangulations. Therefore we first define some local regular and irregular refinement rules that are applied to single elements. The global refinement algorithm then describes how these local rules can be combined and rearranged in order to ensure consistency as well as stability. It is given in a rather general form and includes also grid coarsening. 1991 Mathematics Subject Classifications: 65N50, 65N55 Key words: Tetrahedral grid refinement, stable refinements, consistent triangulations, green closure, grid coarsening. Verfeinerung von TetraederGittern. Es wird ein Verfeinerungsalgorithmus fur unstrukturierte TetraederGitter vorgestellt, der moglicherweise stark nichtuniforme aber dennoch konsistente (d.h. geschlossene) und stabile Triangulierungen liefert. Dazu definieren w...
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 ..."
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Cited by 49 (5 self)
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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.
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.
A nonoverlapping domain decomposition method for Maxwell’s equations in three dimensions
 SIAM J. Numer. Anal
"... Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with muc ..."
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Cited by 35 (10 self)
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Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with much simpler coarse solvers. Though the condition number of the preconditioned system may not have a good bound, we are able to show that the convergence rate of the PCG method with such substructuring preconditioner is nearly optimal, and also robust with respect to the (possibly large) jumps of the coefficient in the elliptic equation. 1.
Some Nonoverlapping Domain Decomposition Methods
, 1998
"... . The purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving secondorder elliptic problems in two and three dimensions. The methods under scrutiny fall into two major categories: the substructuringtype methods and the Neumann ..."
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Cited by 35 (6 self)
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. The purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving secondorder elliptic problems in two and three dimensions. The methods under scrutiny fall into two major categories: the substructuringtype methods and the NeumannNeumanntype methods. The basic framework used for analysis is the parallel subspace correction method or additive Schwarz method, and other technical tools include localglobal and globallocal techniques. The analyses for both two and threedimensional cases are carried out simultaneously. Some internal relationships between various algorithms are observed and several new variants of the algorithms are also derived. Key words. nonoverlapping domain decomposition, Schur complement, localglobal and globallocal techniques, jumps in coe#cients, substructuring, NeumannNeumann, balancing methods AMS subject classifications. 65N30, 65N55, 65F10 PII. S0036144596306800 1. Introduction. T...
Monotone Multigrid Methods for Elliptic Variational Inequalities I
 I. Numer. Math
, 1993
"... . We derive fast solvers for discrete elliptic variational inequalities of the first kind (obstacle problems) as resulting from the approximation of related continuous problems by piecewise linear finite elements. Using basic ideas of successive subspace correction, we modify wellknown relaxation ..."
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Cited by 32 (9 self)
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. We derive fast solvers for discrete elliptic variational inequalities of the first kind (obstacle problems) as resulting from the approximation of related continuous problems by piecewise linear finite elements. Using basic ideas of successive subspace correction, we modify wellknown relaxation methods by extending the set of search directions. Extended underrelaxations are called monotone multigrid methods, if they are quasioptimal in a certain sense. By construction, all monotone multigrid methods are globally convergent. We take a closer look at two natural variants, the standard monotone multigrid method and a truncated version. For the considered model problems, the asymptotic convergence rates resulting from the standard approach suffer from insufficient coarsegrid transport, while the truncated monotone multigrid method provides the same efficiency as in the unconstrained case. Key words: obstacle problems, adaptive finite element methods, multigrid methods AMS (MOS) subje...
Energy Optimization of Algebraic Multigrid Bases
, 1998
"... . We propose a fast iterative method to optimize coarse basis functions in algebraic multigrid by minimizing the sum of their energies, subject to the condition that linear combinations of the basis functions equal to given zero energy modes, and subject to restrictions on the supports of the coarse ..."
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Cited by 27 (2 self)
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. We propose a fast iterative method to optimize coarse basis functions in algebraic multigrid by minimizing the sum of their energies, subject to the condition that linear combinations of the basis functions equal to given zero energy modes, and subject to restrictions on the supports of the coarse basis functions. The convergence rate of the minimization algorithm is bounded independently of the meshsize under usual assumptions on finite elements. The first iteration gives exactly the same basis functions as our earlier method using smoothed aggregation. The construction is presented for scalar problems as well as for linear elasticity. Computational results on difficult industrial problems demonstrate that the use of energy minimal basis functions improves algebraic multigrid performance and yields a more robust multigrid algorithm than smoothed aggregation. 1. Introduction. This paper is concerned with aspects of the design of Algebraic Multigrid Methods (AMG) for the solution of ...