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


Local Multilevel Methods for Second-Order Elliptic Problems with Highly Discontinuous Coefficients

Huangxin Chen 1, Xuejun Xu 2, Weiying Zheng 2

1 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
doi:10.4208/jcm.1109-m3401

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.

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

AMS subject classifications: 65F10, 65N30.


Email: chx@xmu.edu.cn, xxj@lsec.cc.ac.cn, zwy@lsec.cc.ac.cn
 

The Global Science Journal