Hierarchical decomposition of domains with fractures

Gebauer, Susanna; Kornhuber, Ralf; Yserentant, Harry

doi:10.1090/S0025-5718-05-01792-8

Hierarchical decomposition of domains with fractures
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by Susanna Gebauer, Ralf Kornhuber and Harry Yserentant PDF

Math. Comp. 75 (2006), 73-90 Request permission

Abstract:

We consider the efficient and robust numerical solution of elliptic problems with jumping coefficients occurring on a network of thin fractures. We present an iterative solution concept based on a hierarchical separation of the fractures and the surrounding rock matrix. Upper estimates for the convergence rates are independent of the width of the fractures and of the jumps of the coefficients. Inexact solution of the local subproblems is also considered. The theoretical results are illustrated by numerical experiments.

References

Thomas Apel and Joachim Schöberl, Multigrid methods for anisotropic edge refinement, SIAM J. Numer. Anal. 40 (2002), no. 5, 1993–2006. MR 1950630, DOI 10.1137/S0036142900375414
Lori Badea, Xue-Cheng Tai, and Junping Wang, Convergence rate analysis of a multiplicative Schwarz method for variational inequalities, SIAM J. Numer. Anal. 41 (2003), no. 3, 1052–1073. MR 2005195, DOI 10.1137/S0036142901393607
Peter Bastian, Zhangxin Chen, Richard E. Ewing, Rainer Helmig, Hartmut Jakobs, and Volker Reichenberger, Numerical simulation of multiphase flow in fractured porous media, Numerical treatment of multiphase flows in porous media (Beijing, 1999) Lecture Notes in Phys., vol. 552, Springer, Berlin, 2000, pp. 50–68. MR 1876009, DOI 10.1007/3-540-45467-5_{4}
Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275, DOI 10.1007/978-3-642-66451-9
D. Braess, Towards algebraic multigrid for elliptic problems of second order, Computing 55 (1995), no. 4, 379–393 (English, with English and German summaries). MR 1370108, DOI 10.1007/BF02238488
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258, DOI 10.1007/978-1-4757-4338-8
Peter Deuflhard and Reinhard Hochmuth, Multiscale analysis of thermoregulation in the human microvascular system, Math. Methods Appl. Sci. 27 (2004), no. 8, 971–989. MR 2055285, DOI 10.1002/mma.499
Reinhard Hochmuth and Peter Deuflhard, Multiscale analysis for the bio-heat transfer equation—the nonisolated case, Math. Models Methods Appl. Sci. 14 (2004), no. 11, 1621–1634. MR 2103093, DOI 10.1142/S0218202504003775
S. Gebauer. Hierarchische Gebietszerlegung für die gesättigte Grundwasserströmung in Kluftaquifersystemen. Doctoral thesis, Institute for Mathematics II, FU Berlin, 2004.
S. Gebauer, L. Neunhäuserer, R. Kornhuber, S. Ochs, R. Hinkelmann, and R. Helmig. Equidimensional modelling of flow and transport processes in fractured porous systems I. In Hassanizadeh et al., editor, Computational Methods in Water Resources’, pages 335–342. Elsevier, 2002.
M. Heisig, R. Lieckfeldt, G. Wittum, G. Mazurkevich, and G. Lee. Non steady-state descriptions of drug permeation through stratum corneum I: The biphasic brick-and-mortar-model. Pharmaceutical research, 13:421–426, 1996.
R. Helmig. Theorie und Numerik der Mehrphasenströmungen in geklüftet porösen Medien. Ph.D. thesis, Universität Hannover, 1993.
O. Kolditz, J. de Jonge, M. Beinhorn, M. Xie, T. Kalbacher, W. Wang, S. Bauer, C.I. McDermott, R. Kaiser, and M. Kohlmeier. ROCKFLOW - Theory and users manual. Release 3.9. Preprint 2003-37, Center of Applied Geosciences, Geohydrology/HydroInformatics, University of Tübingen, 2003.
L. Neunhäuserer. Diskretisierungsansätze zur Modellierung von Strömungs- und Transportprozessen in geklüftet-porösen Medien. Ph.D. thesis, Institut für Wasserbau, Universität Stuttgart, 2003.
L. Neunhäuserer, S. Gebauer, S. Ochs, R. Hinkelmann, R. Kornhuber, and R. Helmig. Equidimensional modelling of flow and transport processes in fractured porous systems II. In Hassanizadeh et al., editor, Computational Methods in Water Resources’, pages 343–350. Elsevier, 2002.
N. Neuss. Homogenisierung und Mehrgitter. Ph.D. thesis, ICA, Universität Stuttgart, 1996.
Nicolas Neuss, $V$-cycle convergence with unsymmetric smoothers and application to an anisotropic model problem, SIAM J. Numer. Anal. 35 (1998), no. 3, 1201–1212. MR 1619887, DOI 10.1137/S0036142996310848
A. Quarteroni, M. Tuveri, and A. Veneziani. Computational vascular fluid dynamics: problems, models and methods. Comp. Vis. Sci., 2:163–197, 2000.
J. W. Ruge and K. Stüben, Algebraic multigrid, Multigrid methods, Frontiers Appl. Math., vol. 3, SIAM, Philadelphia, PA, 1987, pp. 73–130. MR 972756
A. Silberhorn-Hemminger. Modellierung von Kluftaquifersystemen: Geostatistische Analyse und deterministisch - stochastische Kluftgenerierung. Ph.D. thesis, Institut für Wasserbau, Universität Stuttgart, 2001.
Andrea Toselli and Olof Widlund, Domain decomposition methods—algorithms and theory, Springer Series in Computational Mathematics, vol. 34, Springer-Verlag, Berlin, 2005. MR 2104179, DOI 10.1007/b137868
Alexander Ženíšek and Michèle Vanmaele, The interpolation theorem for narrow quadrilateral isoparametric finite elements, Numer. Math. 72 (1995), no. 1, 123–141. MR 1359711, DOI 10.1007/s002110050163
C. Wagner. Introduction to algebraic multigrid. Course notes, Universität Heidelberg, 1998.
Jinchao Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), no. 4, 581–613. MR 1193013, DOI 10.1137/1034116
Harry Yserentant, On the multilevel splitting of finite element spaces, Numer. Math. 49 (1986), no. 4, 379–412. MR 853662, DOI 10.1007/BF01389538
Harry Yserentant, Old and new convergence proofs for multigrid methods, Acta numerica, 1993, Acta Numer., Cambridge Univ. Press, Cambridge, 1993, pp. 285–326. MR 1224685, DOI 10.1017/S0962492900002385