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

SUMMARY 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...

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Vydáno v:Numerical linear algebra with applications Ročník 21; číslo 1; s. 24 - 38
Hlavní autor: Zhu, Yunrong
Médium: Journal Article
Jazyk:angličtina
Vydáno: Oxford Wiley Subscription Services, Inc 01.01.2014
Témata:
Algorithms
Convergence
Deterioration
Discretization
Eigenvalues
Linear systems
Mathematical models
ISSN:1070-5325, 1099-1506
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Shrnutí:SUMMARY 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. Copyright © 2012 John Wiley & Sons, Ltd.
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ISSN:1070-5325
1099-1506
DOI:10.1002/nla.1856