Local Multilevel Methods for Second-Order Elliptic Problems with Highly Discontinuous Coefficients
Huangxin Chen 1, Xuejun Xu 2, Weiying Zheng 21 School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China
1 LSEC, ICMSEC, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P.O. Box 2719, Beijing, 100190, China
2 LSEC, ICMSEC, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100190, China
Received 2010-4-28 Accepted 2011-9-1
Available online 2012-5-7
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.
Key words: Local multilevel method, Adaptive finite element method, Preconditioned conjugate gradient method, Discontinuous coefficients.
AMS subject classifications: 65F10, 65N30.
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