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Local multilevel preconditioners for elliptic equations with jump coefficients on bisection grids

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Computing and Visualization in Science

Abstract

The goal of this paper is to design optimal multilevel solvers for the finite element approximation of second order linear elliptic problems with piecewise constant coefficients on bisection grids. Local multigrid and BPX preconditioners are constructed based on local smoothing only at the newest vertices and their immediate neighbors. The analysis of eigenvalue distributions for these local multilevel preconditioned systems shows that there are only a fixed number of eigenvalues which are deteriorated by the large jump. The remaining eigenvalues are bounded uniformly with respect to the coefficients and the meshsize. Therefore, the resulting preconditioned conjugate gradient algorithm will converge with an asymptotic rate independent of the coefficients and logarithmically with respect to the meshsize. As a result, the overall computational complexity is nearly optimal.

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Acknowledgments

The first author is supported in part by NSF Grant DMS-0811272, NSF Grant DMS-1115961, and in part by Department of Energy prime award #DE-SC0006903. The second and fourth authors were supported in part by NSF Awards 1065972 and 1217175, by DTRA Award HDTRA-09-1-0036, and by subcontract to DOE Award #DE-SC0006903. The third author was supported in part by NSF DMS 1217142 and DOE Award #DE-SC0006903. The fourth author was supported in part by NSF DMS 1319110 and the University Research Committee Grant No. F119 at Idaho State University, Pocatello, Idaho. This work is also partially supported by the Beijing International Center for Mathematical Research.

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Correspondence to Yunrong Zhu.

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Chen, L., Holst, M., Xu, J. et al. Local multilevel preconditioners for elliptic equations with jump coefficients on bisection grids. Comput. Visual Sci. 15, 271–289 (2012). https://doi.org/10.1007/s00791-013-0213-4

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