P. Van ek, J. Mandel, and M. Brezina, Algebraic Multigrid on Unstructured Meshes, UCD/CCM Report 34, University of Colorado at Denver, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....case is the coarsening process of scalar AMG methods. This process can lead to di erent coarse grids for the di erent physical unknowns, although all unknowns were discretized on the same ( ne) grid. Even worse, coarse grids from scalar AMG coarsening can consist of Smoothed aggregation AMG [21, 22] allows for the prescription of the functions, i.e. vectors, which should be interpolated accurately on coarser grids. only one physical unknown. The second reason is the interpolation of scalar AMG. The quality of interpolation of the kernel elements of the continuous operator plays a ....

P. Van ek, J. Mandel, and M. Brezina, Algebraic Multigrid on Unstructured Meshes, UCD/CCM Report 34, University of Colorado at Denver, 1994.

....to our three unstructured multigrid methods to provide some indication of the performance that we can hope to achieve. Smoothed aggregation multigrid preconditioning can guarantee a condition number of O(log 3 (n) for 3D elasticity and O(log(n) with some assumptions about the problem [38, 37]. This is not optimal but it is close. It is only an upper bound, but our experiments in x6 indicate that in practice we can expect O(1) iterations on elasticity problems. Condition number results for plain aggregation are greater than polylogarithmic in n for 3D problems: O(n 1 3 log(n) ....

P. Vanek, Jan Mandel, and Marian Brezina. Algebraic multigrid on unstructured meshes. UCD/CCM Report 34, Center for Computational Mathematics,University of Colorado at Denver, December 1994.

.... 28, 30, 29] Research continued at a modest pace into the late 1980 s and early 1990 s [18, 14, 21, 25, 20, 26, 22] Recently, however, there has been a major resurgence of interest in the eld, for classical AMG as de ned in [29] as well as for a host of other algebraictype multilevel methods [3, 16, 34, 6, 2, 4, 5, 15, 33, 17, 35, 36, 37]. Largely, this resurgence in AMG research is due to the need to solve increasingly larger systems, with hundreds of millions or billions of unknowns, on unstructured grids. The size of these problems dictates the use of large scale parallel processing, which in turn demands algorithms that scale ....

P. Van ek, J. Mandel, and M. Brezina, Algebraic multigrid on unstructured meshes. UCD/CCM Report 34, Center for Computational Mathematics, University of Colorado at Denver, December 1994. http://www-math.cudenver.edu/ccmreports/rep34.ps.gz.

....in x6 show some indication of super polylogarithmic complexity, but are not conclusive without access to larger test 2 problems and computers. Smoothed aggregation can provide us with a guarantee of O(log 3 (n) iterations [27] and O(log(n) iterations with some assumptions about the problem [28]. Our experiments in x6 indicate O(1) iterations on an elasticity problem with continuum elements with up to 76 million equations with full multigrid (note, full multigrid adds a log(n) term to the parallel complexity) There is a body of unstructured geometric multigrid work that provides ....

P. Vanek, Jan Mandel, and Marian Brezina. Algebraic multigrid on unstructured meshes. UCD/CCM Report 34, Center for Computational Mathematics,University of Colorado at Denver, December 1994.

.... 29, 28] Research continued at a modest pace into the late 1980 s and early 1990 s [18, 14, 20, 24, 19, 25, 21] Recently, however, there has been a major resurgence of interest in the field, for classical AMG as defined in [28] as well as for a host of other algebraictype multilevel methods [3, 16, 33, 6, 2, 4, 5, 15, 32, 17, 34, 35, 36]. Largely, this resurgence in AMG research is due to the need to solve increasingly larger systems, with hundreds of millions or billions of unknowns, on unstructured grids. The size of these problems dictates the use of large scale parallel processing, which in turn demands algorithms that scale ....

P. Van ek, J. Mandel, and M. Brezina, Algebraic multigrid on unstructured meshes. UCD/CCM Report 34, Center for Computational Mathematics, University of Colorado at Denver, December 1994. http://www-math.cudenver.edu/ccmreports/rep34.ps.gz.

....the usage of algebraic multigrid methods. One can trace two main approaches in the development of algebraic multilevel methods: ffl methods using grid structure information [ABDJP81, DJ87, Kuz89, AV89, dZ90, HK91, FG91, Fuh94, Reu95, WKW95] and others ffl methods using only the matrix on input [RS87, VMB94, Bra95] In this paper, a modular algebraic multigrid method is proposed. This method is intended to be able to use grid structure information if such information can be provided, and otherwise, to create it s own coarse grid sequences. It should consist of three main steps: I. Coarsening, symbolic ....

P. Van ek, J. Mandel, and M. Brezina, Algebraic multigrid on unstructured meshes, manuscript, University of Colorado, Denver, CO., 1994. ftp://tiger.denver.colorado.edu/pub/reports.

....author is visiting from University of West Bohemia, Americk a 42, 306 14 Plzen, Czech Republic 1 it has a remarkably low computational complexity since the typical coarsening ratio is about three in each dimension. Almost optimal theoretical bounds for our method were given by the authors in [15] for second order problems and under natural assumptions on the coarse level hierarchy that tend to be satisfied by our coarsening algorithm, namely that the coarsening is by about the factor of three, and that the aggregates of the nodes are based on aggregated elements that form a reasonable ....

....and used to verify the assumptions of the multilevel regularity free approach of Bramble, Pasciak, Wang, and Xu [3] The theory can be extended to fourth order problems once similar energy bounds are available for that case. The part of this paper dealing with second order problems is based on [15]. The algorithm for fourth order problems is new. For more details and theory for the second order case, see [15] For other multigrid approaches to the biharmonic equation, see [5, 9, 16, 8] For a multigrid theory for the biharmonic equation with non nested finite element spaces, see [2] 1.1. ....

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P. Van ek, J. Mandel, and M. Brezina, Algebraic multigrid on unstructured meshes, Tech. Report 34, UCD/CCM, 1994.

....results in a method with good computational complexity and scalability. This method falls under the abstract framework of black box two level iterative methods based on the concept of smoothed aggregations [18] The objective of the smoothing of the coarse basis functions is to reduce their energy [18, 17, 12]. For a related theoretical analysis of such two level methods with high order polynomial smoothing of coarse basis functions, see [3] where a convergence result uniform both with respect to coarse and fine level meshsize was proved for second order eliptic problems discretized on unstructured ....

P. Van ek, J. Mandel, and M. Brezina, Algebraic multigrid on unstructured meshes, UCD/CCM Report 34, Center for Computational Mathematics, University of Colorado at Denver, December 1994.

....is taken by an upper bound on the energy of the basis functions in the grids hierarchy. The classical approach to the design of algebraic multigrid methods [5,8, 23,24] has been to attempt to have the intergrid transfer operators possess approximation properties similar to geometric multigrid. In [29,30], we have proposed an alternative set of objectives, which includes minimization of energy: the basis functions on the coarse levels should have as small energy as possible; preservation of null space: the span of basis functions on each coarse level should contain zero energy modes, at ....

....or, equivalently, the prolongation operators. The objectives were motivated by the multigrid theory of [4] recognition of the need to represent zero energy modes in the coarse space exactly [14,16] and early work on algebraic multigrid with prolongation by smoothed aggregation [25,26] In [6,29 31], we have proposed the prolongation by smoothed aggregation as an attempt to satisfy the above objectives approximately. The energy of the basis functions and convergence of the resulting multigrid method can be further improved by more smoothing of the coarse basis functions, but at the ....

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P. Van ek, J. Mandel, and M. Brezina, Algebraic multigrid on unstructured meshes, UCD/CCM Report 34, Center for Computational Mathematics, University of Colorado at Denver, December 1994. http://wwwmath. cudenver.edu/ccmreports/rep34.ps.gz.

....user input of nodal coordinates to generate linear functions, which made practical treatment of fourth order problems possible. Similarly, rigid body motions are generated to improve convergence for elasticity problems that are ill conditioned due to boundary conditions. In an earlier paper [20], we have developed an analysis of a simplified version of the method without input of linear functions and for second order problems only. The analysis from [20] does not apply here, so we rely on heuristic arguments and practical experience instead. Numerical experiments reported here confirm ....

....are generated to improve convergence for elasticity problems that are ill conditioned due to boundary conditions. In an earlier paper [20] we have developed an analysis of a simplified version of the method without input of linear functions and for second order problems only. The analysis from [20] does not apply here, so we rely on heuristic arguments and practical experience instead. Numerical experiments reported here confirm that the new variant of the method is superior to the old one for the biharmonic equation and for elasticity, particularly for shells. For another approach to ....

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P. Van ek, J. Mandel, and M. Brezina, Algebraic multigrid on unstructured meshes, SIAM J. Sci. Comp., Submitted.

....to apply the theory of [4] in a straightforward manner, one needs to establish that the discrete norms in the artificially constructed coarse spaces are uniformly equivalent to appropriately scaled L 2 norms, and establish the weak approximate property in those norms. We have done this in [27] under additional (though quite reasonable) assumptions on the supports of the coarse basis functions. Essentially, we had to assume that the basis functions in the coarse space hierarchy are associated with a division of the domain into subdomains that behave much like finite elements. Verifying ....

P. Van ek, J. Mandel, and M. Brezina, Algebraic multigrid on unstructured meshes, UCD/CCM Report 34, Center for Computational Mathematics, University of Colorado at Denver, December 1994. http://www-math.cudenver.edu/ccmreports/rep34.ps.gz.

.... has been the focus of interest in the last years, as to which the growing list of references dealing with the topic attests (e.g. the pioneering works leading to AMG of the present day by Brandt, McCormick, Ruge and Stuben [16, 77, 78, 82] and the more recent papers by Vanek, Mandel and Brezina [87, 86]. Our goal will be to design and study such algebraic methods. Focusing on the class of problems arising from the finite element discretization of elliptic partial differential equations, we propose several iterative solvers. Although these are efficient methods in their own right, we will view ....

....freedom per subdomain for scalar second order problems) We present two substructuring methods avoiding the exact subdomain solvers. First of them is an extended BDD. The second one, presented in Chapter 5, is more unorthodox and is closely related to the multigrid method with smoothed aggregation [52, 86, 87]. 4.2 The Inexact Solvers Properties Our aim is to replace the exact solvers with approximate ones. In order to simplify the notation, in this section we describe the inexact solvers and state some of their properties used in our proofs. Assume that all algebraic systems Ax = f (4.1) with a n ....

P. Van ek, J. Mandel, and M. Brezina, Algebraic multigrid on unstructured meshes, UCD/CCM Report 34, Center for Computational Mathematics, University of Colorado at Denver, December 1994. Submitted to SISC.

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