TY - JOUR

T1 - A two-grid SA-AMG convergence bound that improves when increasing the polynomial degree

AU - Hu, Xiaozhe

AU - Vassilevski, Panayot S.

AU - Xu, Jinchao

N1 - Publisher Copyright:
Published 2016. This article is a U.S. Government work and is in the public domain in the USA.

PY - 2016/8/1

Y1 - 2016/8/1

N2 - In this paper, we consider the convergence rate of a smoothed aggregation algebraic multigrid method, which uses a simple polynomial (1 − t)ν or an optimal Chebyshev-like polynomial to construct the smoother and prolongation operator. The result is purely algebraic, whereas a required main weak approximation property of the tentative interpolation operator is verified for a spectral element agglomeration version of the method. More specifically, we prove that, for partial differential equations (PDEs), the two-grid method converges uniformly without any regularity assumptions. Moreover, the convergence rate improves uniformly when the degree of the polynomials used for the smoother and the prolongation increases. Such a result, as is well-known, would imply uniform convergence of the multilevel W-cycle version of the algorithm. Numerical results, for both PDE and non-PDE (graph Laplacian) problems are presented to illustrate the theoretical findings. Published 2016. This article is a U.S. Government work and is in the public domain in the USA.

AB - In this paper, we consider the convergence rate of a smoothed aggregation algebraic multigrid method, which uses a simple polynomial (1 − t)ν or an optimal Chebyshev-like polynomial to construct the smoother and prolongation operator. The result is purely algebraic, whereas a required main weak approximation property of the tentative interpolation operator is verified for a spectral element agglomeration version of the method. More specifically, we prove that, for partial differential equations (PDEs), the two-grid method converges uniformly without any regularity assumptions. Moreover, the convergence rate improves uniformly when the degree of the polynomials used for the smoother and the prolongation increases. Such a result, as is well-known, would imply uniform convergence of the multilevel W-cycle version of the algorithm. Numerical results, for both PDE and non-PDE (graph Laplacian) problems are presented to illustrate the theoretical findings. Published 2016. This article is a U.S. Government work and is in the public domain in the USA.

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U2 - 10.1002/nla.2053

DO - 10.1002/nla.2053

M3 - Article

AN - SCOPUS:84977651671

VL - 23

SP - 746

EP - 771

JO - Numerical Linear Algebra with Applications

JF - Numerical Linear Algebra with Applications

SN - 1070-5325

IS - 4

ER -