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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| Vydané v: | Numerical linear algebra with applications Ročník 21; číslo 1; s. 24 - 38 |
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Oxford
Wiley Subscription Services, Inc
01.01.2014
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| Abstract | 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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| AbstractList | 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 copyright 2012 John Wiley & Sons, Ltd. 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. |
| Author | Zhu, Yunrong |
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| Snippet | SUMMARY In this paper, we present a multigrid V-cycle preconditioner for the linear system arising from piecewise linear nonconforming Crouzeix-Raviart... In this paper, we present a multigrid V-cycle preconditioner for the linear system arising from piecewise linear nonconforming Crouzeix-Raviart discretization... |
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| SubjectTerms | Algorithms Convergence Deterioration Discretization Eigenvalues Linear systems Mathematical models |
| Title | Analysis of a multigrid preconditioner for Crouzeix-Raviart discretization of elliptic partial differential equation with jumpcoefficients |
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