Abstract
In this paper we study some nonoverlapping domain decomposition methods for solving a class of elliptic problems arising from composite materials and flows in porous media which contain many spatial scales. Our preconditioner differs from traditional domain decomposition preconditioners by using a coarse solver which is adaptive to small scale heterogeneous features. While the convergence rate of traditional domain decomposition algorithms using coarse solvers based on linear or polynomial interpolations may deteriorate in the presence of rapid small scale oscillations or high aspect ratios, our preconditioner is applicable to multiple-scale problems without restrictive assumptions and seems to have a convergence rate nearly independent of the aspect ratio within the substructures. A rigorous convergence analysis based on the Schwarz framework is carried out, and we demonstrate the efficiency and robustness of the proposed preconditioner through numerical experiments which include problems with multiple-scale coefficients, as well problems with continuous scales.
Similar content being viewed by others
References
Bramble, J.H., Xu, J. Some Estimates For a Weighted L 2-Projection. Math. Pomp., 1991, 57(195): 463–476
Chan, T., Zou, J. A Convergence Theory of Multilevel Additive Schwarz Methods on Unstructured Meshes. Numerical Algorithms, 1996, 13(3-4): 365–398
Dryja, M., Smith, B.F., Widlund, O.B. Schwarz Analysis of Iterative Substructuring Algorithms for Elliptic Problems in Three Dimensions. SIAM J. Numer. Anal., 1994, 31(6): 1662–1694
Hou, T.Y., Wu, X-H. A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media. J. Comput. Phys., 1997, 134(1): 169–189
Hou, T.Y., Wu, X-H., Cai, Z. Convergence of a Multiscale Finite Element Method for Elliptic Problems with Rapidly Oscillating Coefficients. Math. Comp., 1999, 68(227): 913–943
Jikov, V.V., Kozlov, S.M., Oleinik, O.A. Homogenization of Differential Operators and Integral Functionals. Springer-Verlag, New York, 1994
Mandel, J., Lett, G.S. Domain Decomposition Preconditioning for p-version Finite Elements with High Aspect Ratios. Applied Numerical Analysis, 1991, 8: 411–425
Quarteroni, A., Valli, A. Domain Decomposition Methods for Partial Differential Equations. Oxford University Press, Oxford, 1999
Renard P., de Marsily, G. Calculating Equivalent Permeability. Advances in Water Resources, 1997, 20: 253–278
Sarkis, M. Nonstandard Coarse Spaces and Schwarz Methods for Elliptic Problems with Discontinuous Coefficients using Non-Conforming Elements. Numer. Math., 1997, 77: 383–406
Smith, B.F., Bjørstad, P., Gropp, W. Domain Decomposition Methods for Partial Differential Equations. Cambridge University Press, Cambridge, 1996
Smith, B.F. An Optimal Domain Decomposition Preconditioner for the Finite Element Solution of Linear Elasticity Problems. SIAM J. Sci. Statist. Comput., 1992, 13(1): 364–378
Tai, X-C., Espedal, M.S. Rate of Convergence of Some Space Decomposition Methods for Linear and Nonlinear Problems. SIAM J. Numer. Anal., 1998, 35(4): 1558–1570
Tai, X-C., Xu, J. Global and Uniform Convergence of Subspace Correction Methods for some Convex Optimization Problems. Math. Comp., 2002, 71(237): 105–124
Xu, J. Iterative Methods by Space Decomposition and Subspace Correction. SIAM Rev., 1992, 34: 581–613
Xu, J., Zou, J. Some Nonoverlapping Domain Decomposition Methods. SIAM Rev., 1998, 40(4): 857–914
Author information
Authors and Affiliations
Corresponding author
Additional information
* Supported by STATOIL under the VISTA program;
** Supported in part by a grant from National Science Foundation under the contract DMS-0073916, and by a grant from Army Research Office under the contract DAAD19-99-1-0141.
Rights and permissions
About this article
Cite this article
Aarnes*, J., Hou**, T.Y. Multiscale Domain Decomposition Methods for Elliptic Problems with High Aspect Ratios. Acta Mathematicae Applicatae Sinica, English Series 18, 63–76 (2002). https://doi.org/10.1007/s102550200004
Received:
Issue Date:
DOI: https://doi.org/10.1007/s102550200004