Some Remarks on a Multigrid Preconditioner

Xu, Jinchao; Qin, Jinshui

doi:10.1137/0915012

Cited by 12 publications

(8 citation statements)

References 9 publications

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“…However, if one needs to solve (1.1) for a large number (M) of different right-hand sides g (M n, which would happen if one tries to optimize specific properties of u with respect to g) then the methods proposed here have a practical use since they basically say that after solving (1.1) n times it is sufficient to solve (1.1) on a coarse mesh (with N 0.01 nodes instead of N , for instance) M times. Let us recall that (1.1) can be solved in O(N (ln N ) n+3 ) operations using hierarchical matrix methods [9,10,11,12,13] or in O(N ) operations using iterative methods (see [76,77]). …”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Metric‐based upscaling

Owhadi

Zhang

2006

Comm Pure Appl Math

199

184

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We consider divergence form elliptic operators in dimension n ≥ 2 with L ∞ coefficients. Although solutions of these operators are only Hölder-continuous, we show that they are differentiable (C 1,α ) with respect to harmonic coordinates. It follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales. This new numerical homogenization method is based on the transfer of a new metric in addition to traditional averaged (homogenized) quantities from subgrid scales into computational scales. Error bounds can be given and this method can also be used as a compression tool for differential operators.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Metric‐based upscaling

Owhadi

Zhang

2006

Comm Pure Appl Math

199

184

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“…In the first article [2] of the trilogy, it was shown that the BPX preconditioner was optimal on the meshes under the local 2D and 2D red-green, as well as local 2D and 3D red, refinement procedures. The classical BPX preconditioner [8,20] can be written as an action of the operator X as follows:…”

Section: Overview Of the Multilevel Methodsmentioning

confidence: 99%

An Odyssey into Local Refinement and Multilevel Preconditioning III: Implementation and Numerical Experiments

Aksoylu¹,

Bond²,

Holst³

2003

SIAM J. Sci. Comput.

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Abstract. In this article, we examine the Bramble-Pasciak-Xu (BPX) preconditioner in the setting of local 2D and 3D mesh refinement. While the available optimality results for the BPX preconditioner have been constructed primarily in the setting of uniformly refined meshes, a notable exception is the 2D result due to Dahmen and Kunoth, which established BPX optimality on meshes produced by a restricted class of local 2D red-green refinement. The purpose of this article is to extend the original 2D Dahmen-Kunoth result to several additional types of local 2D and 3D red-green (conforming) and red (non-conforming) refinement procedures. The extensions are accomplished through a 3D extension of the 2D framework in the original Dahmen-Kunoth work, by which the question of optimality is reduced to establishing that locally enriched finite element subspaces allow for the construction of a scaled basis which is formally Riesz stable. This construction in turn rests entirely on establishing a number of geometrical properties between neighboring simplices produced by the local refinement algorithms. These properties are then used to build Riesz-stable scaled bases for use in the BPX optimality framework. Since the theoretical framework supports arbitrary spatial dimension d ≥ 1, we indicate clearly which geometrical properties, established here for several 2D and 3D local refinement procedures, must be re-established to show BPX optimality for spatial dimension d ≥ 4. Finally, we also present a simple alternative optimality proof of the BPX preconditioner on quasiuniform meshes in two and three spatial dimensions, through the use of K-functionals and H sstability of L 2 -projection for s ≥ 1. The proof techniques we use are quite general; in particular, the results require no smoothness assumptions on the PDE coefficients beyond those required for well-posedness in H 1 , and the refinement procedures may produce nonconforming meshes.Key words. finite element approximation theory, multilevel preconditioning, BPX, two and three dimensions, local mesh refinement, red and red-green refinement.AMS subject classifications. 65M55, 65N55, 65N22, 65F101. Introduction. In this article, we analyze the impact of local adaptive mesh refinement on the stability of multilevel finite element spaces and on the optimality (linear space and time complexity) of multilevel preconditioners. Adaptive refinement techniques have become a crucial tool for many applications, and access to optimal or near-optimal multilevel preconditioners for locally refined mesh situations is of primary concern to computational scientists. The preconditioners which can be expected to have somewhat favorable space and time complexity in such local refinement scenarios are the hierarchical basis (HB) method, the Bramble-PasciakXu (BPX) preconditioner, and the wavelet modified (or stabilized) hierarchical basis (WHB) method. While there are optimality results for both the BPX and WHB preconditioners in the literature, these are primarily for quasiuniform meshes and/o...

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“…We decided to show details of a general additive multilevel Schwarz (or BPX) preconditioner to emphasize the simplicity of implementation when dealing with mixed methods preconditioning. More details on implementing the matrix action of multilevel preconditioners (including BPX) can be found in [19].…”

Section: With a Suitable Preconditioner P H Replacing A −1mentioning

confidence: 99%

Saddle point least squares preconditioning of mixed methods

Bǎcuţǎ

Jacavage

2019

Computers & Mathematics with Applications

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We present a simple way to discretize and precondition mixed variational formulations. Our theory connects with, and takes advantage of, the classical theory of symmetric saddle point problems and the theory of preconditioning symmetric positive definite operators. Efficient iterative processes for solving the discrete mixed formulations are proposed and choices for discrete spaces that are always compatible are provided. For the proposed discrete spaces and solvers, a basis is needed only for the test spaces and assembly of a global saddle point system is avoided. We prove sharp approximation properties for the discretization and iteration errors and also provide a sharp estimate for the convergence rate of the proposed algorithm in terms of the condition number of the elliptic preconditioner and the discrete inf − sup and sup − sup constants of the pair of discrete spaces.2000 Mathematics Subject Classification. 74S05, 74B05, 65N22, 65N55.

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Some Remarks on a Multigrid Preconditioner

Cited by 12 publications

References 9 publications

Metric‐based upscaling

Metric‐based upscaling

An Odyssey into Local Refinement and Multilevel Preconditioning III: Implementation and Numerical Experiments

Saddle point least squares preconditioning of mixed methods

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