Uniform Convergent Multigrid Methods for Elliptic Problems With Strongly Discontinuous Coefficients

Xu, Jinchao; Zhu, Yunrong

doi:10.1142/s0218202508002619

Cited by 116 publications

(121 citation statements)

References 34 publications

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“…In Section §3, using the method of subspace correction framework [18,91], we discuss the classical V-cycle multigrid method and the BPX preconditioner. We also include a recent result by Xu and Zhu [93] that demonstrates that the conjugate gradient method with classical V-cycle multigrid or BPX-preconditioner as preconditioners provides a robust method with respect to jump discontinuities of coefficients.…”

Section: Multilevel Methods On Quasi-uniform Gridsmentioning

confidence: 99%

“…Recently, Xu and Zhu [93] proved that BPX and multigrid V -cycle lead to a nearly uniform convergent preconditioned conjugate gradient method (see [97] for a similar result on DD preconditioners). We now report this result.…”

Section: Systems With Strongly Discontinuous Coefficientsmentioning

confidence: 96%

“…The BPX and overlapping domain decomposition preconditioners are proven to be robust for some special cases: interface has no cross points [20,66]; every subdomain touches part of the Dirichlet boundary [93]; and quasi-monotone coefficients [33,34]. If the number of levels is fixed, multigrid converges uniformly with the convergence rate ρ k ≤ 1 − δ k where δ ∈ (0, 1) is a constant and k is the number of levels.…”

Section: Systems With Strongly Discontinuous Coefficientsmentioning

confidence: 99%

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Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids

Chen

Nochetto

2009

Multiscale, Nonlinear and Adaptive Approximation

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We give an overview of multilevel methods, such as V-cycle multigrid and BPX preconditioner, for solving various partial differential equations (including H(grad), H(curl) and H(div) systems) on quasi-uniform meshes and extend them to graded meshes and completely unstructured grids. We first discuss the classical multigrid theory on the basis of the method of subspace correction of Xu and a key identity of Xu and Zikatanov. We next extend the classical multilevel methods in H(grad) to graded bisection grids upon employing the decomposition of bisection grids of Chen, Nochetto, and Xu. We finally discuss a class of multilevel preconditioners developed by Hiptmair and Xu for problems discretized on unstructured grids and extend them to H(curl) and H(div) systems over graded bisection grids.

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Section: Multilevel Methods On Quasi-uniform Gridsmentioning

confidence: 99%

Section: Systems With Strongly Discontinuous Coefficientsmentioning

confidence: 96%

Section: Systems With Strongly Discontinuous Coefficientsmentioning

confidence: 99%

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Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids

Chen

Nochetto

2009

Multiscale, Nonlinear and Adaptive Approximation

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“…The resolved cases in Figures 1(a-b) have already been studied extensively, see e.g. [4,3,10,16,25,21,22,24]. …”

Section: Theorem 1 If There Exists An Operatormentioning

confidence: 99%

“…A certain amount of robustness can be recovered for standard piecewise linear coarse spaces by using the multilevel solver as a preconditioner within CG (e.g. [24]). The key tool in all these analyses is the weighted L 2 -projection of Bramble and Xu [1].…”

Section: Introductionmentioning

confidence: 99%

Robust Coarsening in Multiscale PDEs

Scheichl

2013

Lecture Notes in Computational Science and Engineering

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Analysis of a multigrid preconditioner for Crouzeix-Raviart discretization of elliptic partial differential equation with jump coefficients

Zhu

2012

Numer. Linear Algebra Appl.

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In this paper, we present a multigrid V -cycle preconditioner for the linear system arising from piecewise linear nonconforming Crouzeix-Raviart discretization of second-order elliptic problems with jump coefficients. The preconditioner uses standard conforming subspaces as coarse spaces. We showed that the convergence rates of the (multiplicative) two-grid and multigrid V -cycle algorithms will deteriorate rapidly because of large jumps in coefficient. However, the preconditioned systems have only a fixed number of small eigenvalues depending on the large jump in coefficient, and the effective condition numbers are independent of the coefficient and bounded logarithmically with respect to the mesh size. As a result, the two-grid or multigrid preconditioned conjugate gradient algorithm converges nearly uniformly. We also comment on some major differences of the convergence theory between the nonconforming case and the standard conforming case. Numerical experiments support the theoretical results.Here,/ is an open polygonal domain and f 2 L 2 . /. We assume that the diffusion coefficientfor m ¤ n. Developing efficient solvers/preconditioners for the CR discretization is not only important for its own sake, but it has several important applications. In particular, it can be used to develop efficient solvers for other discretizations of (1.1). For example, by using the equivalence between CR discretization and mixed methods (cf. [1]), the preconditioners for CR discretization can be applied to solve mixed finite element discretizations for (1.1) (see [2,3] for the multigrid algorithms in the case of smooth coefficients). This relationship is also used in [4] to analyze multigrid algorithms for mixed finite element discretization on adaptively refined meshes. Another important application *Correspondence to: Yunrong Zhu, Physical Sciences 318 P. O. Box 8085, 25 is on the design of efficient multilevel solvers for interior penalty discontinuous Galerkin (IPDG) methods; we refer to [5] for the Poisson problem, and to [6] for (1.1) with jump coefficients.A lot of work has been done to develop multilevel solvers and preconditioners for the piecewise linear CR discretization of (1.1) in the context of smooth (or constant) coefficients. Multigrid algorithms were developed and analyzed in [7][8][9][10][11], and two-level additive Schwarz preconditioners are presented in [12]. Hierarchical and Bramble-Pasciak-Xu (BPX)-type multilevel preconditioners were proposed in [13], and their optimality was shown in [14]. Most of the aforementioned works define the multilevel structure using the natural sequences of nonconforming spaces. Because these nonconforming spaces are non-nested, special intergrid transfer operators (restriction and prolongation) are needed in the analysis and implementation of the algorithms. Recently, the robustness of the BPX preconditioner using the non-nested nonconforming coarse spaces was shown in [15] for the jump coefficient problem (1.1).On the other hand, we observe that the standard piecewise linea...

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Uniform Convergent Multigrid Methods for Elliptic Problems With Strongly Discontinuous Coefficients

Cited by 116 publications

References 34 publications

Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids

Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids

Robust Coarsening in Multiscale PDEs

Analysis of a multigrid preconditioner for Crouzeix-Raviart discretization of elliptic partial differential equation with jump coefficients

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