Numerical analysis of a multigrid method for spectral approximations

Maday, Yvon; Munoz, R Raul Pagan

doi:10.1007/3-540-51048-6_61

Cited by 8 publications

(7 citation statements)

References 10 publications

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“…The early results on 1-D problems are encouraging and show optimal solver performance [13,14]. However, the extension to higher dimensions results in non-optimal performance due to the dependence of the convergence factor, q, of the multigrid iteration using a Jacobi smoother on the order of the polynomial basis, p. More precisely, q = 1Àc/p, where c is a constant independent of p [13]. It was shown that using a Chebyshev acceleration scheme resulted in the convergence factor being 1 À c= ffiffi ffi p p , but still not independent of p.…”

Section: Introductionmentioning

confidence: 70%

“…Once the polynomial order was reduced to p = 1, the traditional geometric multigrid algorithm could then be used. The early results on 1-D problems are encouraging and show optimal solver performance [13,14]. However, the extension to higher dimensions results in non-optimal performance due to the dependence of the convergence factor, q, of the multigrid iteration using a Jacobi smoother on the order of the polynomial basis, p. More precisely, q = 1Àc/p, where c is a constant independent of p [13].…”

Section: Introductionmentioning

confidence: 94%

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Algebraic multigrid for higher-order finite elements

Heys

Manteuffel

McCormick

et al. 2005

Journal of Computational Physics

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Section: Introductionmentioning

confidence: 70%

Section: Introductionmentioning

confidence: 94%

Algebraic multigrid for higher-order finite elements

Heys

Manteuffel

McCormick

et al. 2005

Journal of Computational Physics

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“…Since V 1 h ⊂ V 2 h , by using (11), (13) and the compact support properties of {w i } i∈S a , we have…”

Section: Two-level Methods Based On Geometric Considerationmentioning

confidence: 99%

A geometric‐based algebraic multigrid method for higher‐order finite element equations in two‐dimensional linear elasticity

Xiao

Shu

Zhao

2008

Numerical Linear Algebra App

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In this paper, we will discuss the geometric-based algebraic multigrid (AMG) method for two-dimensional linear elasticity problems discretized using quadratic and cubic elements. First, a two-level method is proposed by analyzing the relationship between the linear finite element space and higher-order finite element space. And then a geometric-based AMG method is obtained with the existing solver used as a solver on the first coarse level. The resulting AMG method is applied to some typical elasticity problems including the plane strain problem with jumps in Young's modulus. The results of various numerical experiments show that the proposed AMG method is much more robust and efficient than a classical AMG solver that is applied directly to the high-order systems alone. Moreover, we present the corresponding theoretical analysis for the convergence of the proposed AMG algorithms. These theoretical results are also confirmed by some numerical tests.

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“…A year later, a paper by Maday and Muñoz [20] gave the convergence theory for the spectral element multigrid method. They proved that the convergence rate is independent of p in 1D, but in 2D it is O(1−c/ p) for some constant c. Other presentations of work by Rønquist, Patera, Maday, and Muñoz can be found in [21][22][23].…”

Section: Historymentioning

confidence: 99%

The hp‐multigrid method applied to hp‐adaptive refinement of triangular grids

Mitchell

2010

Numerical Linear Algebra App

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SUMMARYRecently the hp version of the finite element method, in which adaptivity occurs in both the size, h, of the elements and in the order, p, of the approximating piecewise polynomials, has received increasing attention. It is desirable to combine this optimal order discretization method with an optimal order algebraic solution method like multigrid. An intriguing notion is to use the values of p as the levels of a multilevel method. In this paper we present such a method, known as hp-multigrid, for high-order finite elements and hp-adaptive grids. We present a survey of the development of p-multigrid and hp-multigrid, define an hp-multigrid algorithm based on the p-hierarchical basis for the p levels and h-hierarchical basis for an h-multigrid solution of the p = 1 'coarse grid' equations, and present numerical convergence results using hp-adaptive grids. The numerical results suggest that the method has a convergence rate of

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Numerical analysis of a multigrid method for spectral approximations

Cited by 8 publications

References 10 publications

Algebraic multigrid for higher-order finite elements

Algebraic multigrid for higher-order finite elements

A geometric‐based algebraic multigrid method for higher‐order finite element equations in two‐dimensional linear elasticity

The hp‐multigrid method applied to hp‐adaptive refinement of triangular grids

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