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

SUMMARYIn 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 s...

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Bibliographic Details
Published in:Numerical linear algebra with applications Vol. 21; no. 1; pp. 24 - 38
Main Author: Zhu, Yunrong
Format: Journal Article
Language:English
Published: Blackwell Publishing Ltd 01.01.2014
Subjects:
conjugate gradient
Crouzeix-Raviart
effective condition number
jump coefficients
multigrid
preconditioner
two-grid
ISSN:1070-5325, 1099-1506
Online Access:Get full text
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Summary:SUMMARYIn 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.
Bibliography:ArticleID:NLA1856
istex:5BE89BF6A594BE4AC1C648868E6315B9D4D0418E
NSF - No. DMS-0715146
DTRA Award - No. HDTRA- 09-1-0036
ark:/67375/WNG-FLT05FT5-0
ISSN:1070-5325
1099-1506
DOI:10.1002/nla.1856