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Volume 30, Issue 3
Local Multilevel Methods for Second-Order Elliptic Problems with Highly Discontinuous Coefficients

Huangxin Chen, Xuejun Xu & Weiying Zheng

DOI: 10.4208/jcm.1109-m3401

J. Comp. Math., 30 (2012), pp. 223-248.

Published online: 2012-06

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  • Abstract

In this paper, local multiplicative and additive multilevel methods on adaptively refined meshes are considered for second-order elliptic problems with highly discontinuous coefficients. For the multilevel-preconditioned system, we study the distribution of its spectrum by using the abstract Schwarz theory. It is proved that, except for a few small eigenvalues, the spectrum of the preconditioned system is bounded quasi-uniformly with respect to the jumps of the coefficient and the mesh sizes. The convergence rate of multilevel-preconditioned conjugate gradient methods is shown to be quasi-optimal regarding the jumps and the meshes. Numerical experiments are presented to illustrate the theoretical findings.

  • Keywords

Local multilevel method, Adaptive finite element method, Preconditioned conjugate gradient method, Discontinuous coefficients.

  • AMS Subject Headings

65F10, 65N30.

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{JCM-30-223, author = {Huangxin Chen, Xuejun Xu and Weiying Zheng}, title = {Local Multilevel Methods for Second-Order Elliptic Problems with Highly Discontinuous Coefficients}, journal = {Journal of Computational Mathematics}, year = {2012}, volume = {30}, number = {3}, pages = {223--248}, abstract = {

In this paper, local multiplicative and additive multilevel methods on adaptively refined meshes are considered for second-order elliptic problems with highly discontinuous coefficients. For the multilevel-preconditioned system, we study the distribution of its spectrum by using the abstract Schwarz theory. It is proved that, except for a few small eigenvalues, the spectrum of the preconditioned system is bounded quasi-uniformly with respect to the jumps of the coefficient and the mesh sizes. The convergence rate of multilevel-preconditioned conjugate gradient methods is shown to be quasi-optimal regarding the jumps and the meshes. Numerical experiments are presented to illustrate the theoretical findings.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1109-m3401}, url = {http://global-sci.org/intro/article_detail/jcm/8427.html} }
TY - JOUR T1 - Local Multilevel Methods for Second-Order Elliptic Problems with Highly Discontinuous Coefficients AU - Huangxin Chen, Xuejun Xu & Weiying Zheng JO - Journal of Computational Mathematics VL - 3 SP - 223 EP - 248 PY - 2012 DA - 2012/06 SN - 30 DO - http://doi.org/10.4208/jcm.1109-m3401 UR - https://global-sci.org/intro/article_detail/jcm/8427.html KW - Local multilevel method, Adaptive finite element method, Preconditioned conjugate gradient method, Discontinuous coefficients. AB -

In this paper, local multiplicative and additive multilevel methods on adaptively refined meshes are considered for second-order elliptic problems with highly discontinuous coefficients. For the multilevel-preconditioned system, we study the distribution of its spectrum by using the abstract Schwarz theory. It is proved that, except for a few small eigenvalues, the spectrum of the preconditioned system is bounded quasi-uniformly with respect to the jumps of the coefficient and the mesh sizes. The convergence rate of multilevel-preconditioned conjugate gradient methods is shown to be quasi-optimal regarding the jumps and the meshes. Numerical experiments are presented to illustrate the theoretical findings.

Huangxin Chen, Xuejun Xu and Weiying Zheng. (2012). Local Multilevel Methods for Second-Order Elliptic Problems with Highly Discontinuous Coefficients. Journal of Computational Mathematics. 30 (3). 223-248. doi:10.4208/jcm.1109-m3401
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