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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 ..."
Abstract

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 ...
Sharp estimates for multigrid rates of convergence with general smoothing and acceleration
 SIAMJ.Numer.Anal.22 (1985), 617–633. MR 87j:65037
"... In this paper, we prove the convergence of the multilevel iterative method for solving linear equations that arise from elliptic partial differential equations. Our theory is presented entirely in terms of the generalized condition number K of the matrix A and the smoothing matrix B. This leads to a ..."
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Cited by 46 (18 self)
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In this paper, we prove the convergence of the multilevel iterative method for solving linear equations that arise from elliptic partial differential equations. Our theory is presented entirely in terms of the generalized condition number K of the matrix A and the smoothing matrix B. This leads to a completely algebraic analysis of the method as an iterative technique for solving linear equations; the properties of the elliptic equation and discretization procedure enter only when we seek to estimate K, just as in the case of most standard iterative methods. Here we consider the fundamental twolevel iteration, and the V and W cycles of the jlevel iteration (j> 2). We prove that the V and W cycles converge even when only one smoothing iteration is used. We present several examples of the computation of using both Fourier analysis and standard finite element techniques. We compare the predictions of our theorems with the actual rate of convergence. Our analysis also shows that accelerated iterative methods, both fixed (Chebyshev) and adaptive (conjugate gradients and conjugate residuals), are effective as smoothing procedures.
Parallel Multigrid in an Adaptive PDE Solver Based on Hashing and SpaceFilling Curves
, 1997
"... this paper is organized as follows: In section 2 we discuss data structures for adaptive PDE solvers. Here, we suggest to use hash tables instead of the usually employed tree type data structures. Then, in section 3 we discuss the main features of the sequential adaptive multilevel solver. Section 4 ..."
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Cited by 39 (3 self)
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this paper is organized as follows: In section 2 we discuss data structures for adaptive PDE solvers. Here, we suggest to use hash tables instead of the usually employed tree type data structures. Then, in section 3 we discuss the main features of the sequential adaptive multilevel solver. Section 4 deals with the partitioning and distribution of adaptive grids with spacefilling curves and section 5 gives the main features of our new parallelized adaptive multilevel solver. In section 6 we present the results of numerical experiments on a parallel cluster computer with up to 64 processors. It is shown that our approach works nicely also for problems with severe singularities which result in locally refined meshes. Here, the work overhead for load balancing and data distribution remains only a small fraction of the overall work load. 2. DATA STRUCTURES FOR ADAPTIVE PDE SOLVERS 2.1. Adaptive Cycle
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...
Algebraic Multigrid On Unstructured Meshes
, 1994
"... An algebraic multigrid algorithm is developed based on prolongations by smoothed aggregation. Coarse levels are generated automatically. Heuristic principles to guide the choice of the coarseninng are introduced. Almost optimal convergence bounds are proved for uniformly elliptic problems, shape re ..."
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Cited by 32 (6 self)
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An algebraic multigrid algorithm is developed based on prolongations by smoothed aggregation. Coarse levels are generated automatically. Heuristic principles to guide the choice of the coarseninng are introduced. Almost optimal convergence bounds are proved for uniformly elliptic problems, shape regular triangulations on the finest level, and a general class of coarse problem hierarchy. Coarsening is by the factor of about three, which guarantees low complexity and bounded energy of the coarse shape functions. Numerical experiments confirm the theory and demonstrate that the method performs well also on a general class of problems with highly variable coefficients and strong anisotropies.
Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences
, 1998
"... We present a multilevel approach for the solution of partial differential equations. It is based on a multiscale basis which is constructed from a onedimensional multiscale basis by the tensor product approach. Together with the use of hash tables as data structure, this allows in a simple way for a ..."
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Cited by 29 (15 self)
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We present a multilevel approach for the solution of partial differential equations. It is based on a multiscale basis which is constructed from a onedimensional multiscale basis by the tensor product approach. Together with the use of hash tables as data structure, this allows in a simple way for adaptive refinement and is, due to the tensor product approach, well suited for higher dimensional problems. Also, the adaptive treatment of partial differential equations, the discretization (involving finite differences) and the solution (here by preconditioned BiCG) can be programmed easily. We describe the basic features of the method, discuss the discretization, the solution and the refinement procedures and report on the results of different numerical experiments.
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 ...
Parallel Adaptive Subspace Correction Schemes with Applications to Elasticity
 Comput. Methods Appl. Mech. Engrg
, 1999
"... : In this paper, we give a survey on the three main aspects of the efficient treatment of PDEs, i.e. adaptive discretization, multilevel solution and parallelization. We emphasize the abstract approach of subspace correction schemes and summarize its convergence theory. Then, we give the main featur ..."
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Cited by 8 (4 self)
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: In this paper, we give a survey on the three main aspects of the efficient treatment of PDEs, i.e. adaptive discretization, multilevel solution and parallelization. We emphasize the abstract approach of subspace correction schemes and summarize its convergence theory. Then, we give the main features of each of the three distinct topics and treat the historical background and modern developments. Furthermore, we demonstrate how all three ingredients can be put together to give an adaptive and parallel multilevel approach for the solution of elliptic PDEs and especially of linear elasticity problems. We report on numerical experiments for the adaptive parallel multilevel solution of some test problems, namely the Poisson equation and Lam'e's equation. Here, we emphasize the parallel efficiency of the adaptive code even for simple test problems with little work to distribute, which is achieved through hash storage techniques and spacefilling curves. Keywords: subspace correction, iter...
On the Convergence of Cascadic Iterations for Elliptic Problems
 Preprint SC 948, KonradZuseZentrum fur Informationstechnik
, 1994
"... . We consider nested iterations, in which the multigrid method is replaced by some simple basic iteration procedure, and call them cascadic iterations. They were introduced by Deuflhard, who used the conjugate gradient method as basic iteration (CCG method). He demonstrated by numerical experiments ..."
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Cited by 7 (0 self)
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. We consider nested iterations, in which the multigrid method is replaced by some simple basic iteration procedure, and call them cascadic iterations. They were introduced by Deuflhard, who used the conjugate gradient method as basic iteration (CCG method). He demonstrated by numerical experiments that the CCG method works within a few iterations if the linear systems on coarser triangulations are solved accurately enough. Shaidurov subsequently proved multigrid complexity for the CCG method in the case of H 2 regular twodimensional problems with quasiuniform triangulations. We show that his result still holds true for a large class of smoothing iterations as basic iteration procedure in the case of two and threedimensional H 1+ff regular problems. Moreover we show how to use cascadic iterations in adaptive codes and give in particular a new termination criterion for the CCG method. Key Words. Finite element approximation, cascadic iteration, nested iteration, smoothing ite...