An analysis of a FETI-DP algorithm on irregular subdomains in the plane

SIAM Journal on Numerical Analysis

Volumn 46, Issue 5, 2008, Pages 2484-2504

  Klawonn, Axel ^a   Rheinbach, Oliver ^a   Widlund, Olof B ^b  

^a UNIVERSITY OF DUISBURG ESSEN   (Germany)

^b NEW YORK UNIVERSITY   (United States)

Author keywords

Domain decomposition; Dual primal FETI; Fractal subdomains; Iterative substructures; John and uniform domains; Preconditioned

Indexed keywords

DOMAIN DECOMPOSITION; DUAL-PRIMAL FETI; ITERATIVE SUBSTRUCTURES; JOHN AND UNIFORM DOMAINS; PRECONDITIONED;

CONJUGATE GRADIENT METHOD; DOMAIN DECOMPOSITION METHODS; DYNAMIC POSITIONING; NUMBER THEORY;

PERCOLATION (SOLID STATE);

EID: 55349129284     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/070688675     Document Type: Article

References (37)

1. G. ACOSTA, R. G. DURAN, AND M. A. MUSOHIETTI, Solutions of the divergence operator on John domains, Adv. Math., 206 (2006), pp. 373-401.
2. G. P. ASTRAKHATSEV, Method of fictitious domains for a second-order elliptic equation with natural boundary conditions, U.S.S.R. Computational Math. and Math. Phys., 18 (1978), pp. 114-121.
3. S. DALAY, K. BUSCHELMAN, V. EIJKHOUT, W. D. GROPP, D. KAUSHIK, M. G. KNEPLEY, L. C. MCINNES, B. F. SMITH, AND H. ZHANG, PETSc users manual, Tech. Report ANL-95/11-Revision 2.1.5, Argonne National Laboratory, Argonne, IL, 2004.
4. S. BALAY, K. BUSHELMAN, W. D. GROUP, D. KAUSHIK, M. G. KNEPLEY, L. C. MCINNES, B. F. SMITH, AND H. ZHANCS, PETSc Web page, available online at http://www.mcs.anl. gov/petsc, 2001.
5. S. BALAY, W. D. GROPP, L. C. MCINNES, AND B. F. SMITH, Efficient management of parallelism in object oriented numerical software libraries, in Modern Software Tools in Scientific Computing, E. Arge, A. M. Bruaset, and H. P. Langtangen, eds., Birkhäuser Press, Basel, Switzerland, 1997, pp. 163-202.
6. B. BOJAHSKI, Remarks on Sobolev imbedding inequalities, in Complex Analysis, Joensuu 1987, Lecture Notes in Math. 1351, Springer, Berlin, 1988, pp. 52-68.
7. J. H. BRAMBLE, J. E. PASCIAK, AND A. H. SCHATZ, The construction of preconditioned for elliptic problems by substructuring. I, Math. Comp., 47 (1986), pp. 103-134.
8. S. C. BRENNER AND L. R. SCOTT, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics 15, 2nd ed., Springer-Verlag, New York, 2002.
9. S. BUCKLEY AND P. KOSKELA, Sobolev-Poincaré implies John, Math. Res. Lett., 2 (1995), pp. 577-593.
10. P. G. CIARLET AND P.-A. RAVIART, Maximum principle and uniform convergence for the finite element method, Comput. Methods Appl. Mech. Engrg., 2 (1973), pp. 17-31.
11. P. CLÉMENT, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. RAIRO Analyse Numérique, 9 (1975), pp. 77-84.
12. C. R. DOHRMANN, A. KLAWONN, AND O. B. WIDLUND, Domain decomposition for less regular subdomains: Overlapping Schwarz in two dimensions, SIAM J. Numer. Anal., 46 (2008), pp. 2153-2168.
13. C. R. DOHRMANN, A. KLAWONN, AND O. B. WIDLUND, Extending theory for domain decomposition algorithms to irregular subdomains, in Proceedings of the 17th International Conference on Domain Decomposition Methods in Science and Engineering, Strobl, Austria, 2006, U. Langer, M. Discacciati, D. Keyes, O. B. Widlund, and W. Zulehner, eds., Lect. Notes Comput. Sei. Engrg. 60, Springer-Verlag, 2007, pp. 255-261.
14. C. R. DOHRMANN, A. KLAWONN, AND O. D. WIDLUND, A family of energy minimizing coarse spaces for overlapping Schwarz preconditioners, in Proceedings of the 17th International Conference on Domain Decomposition Methods in Science and Engineering, Strobl, Austria, 2006, U. Langer, M. Discacciati, D. Keyes, O. B. Widlund, and W. Zulehner, eds., Lect. Notes Comput. Sei. Engrg. 60, Springer-Verlag, 2007, pp. 247-254.
15. M. DRYJA AND O. B. WIDLUND, Some domain decomposition algorithms for elliptic problems, in Iterative Methods for Large Linear Systems, L. Hayes and D. Kincaid, eds., Academic Press, 1990, pp. 273-291.
16. R. G. DURÁN AND M. A. MUSCHIETTI, The Korn inequality for Jones domains, Electron. J. Differential Equations, 2004 (2004), pp. 1-10.
17. C. FARHAT, M. LESOINNE, P. LE TALLEO, K. PIERSON, AND D. RIXEN, FETI-DP: A dualprimal unified FETI method - Part I: A faster alternative to the two-level FETI method, Internat. J. Numer. Methods Engrg., 50 (2001), pp. 1523-1544.
18. H. FEDERER AND W. H. FLEMING, Normal and integral currents, Ann. of Math. (2), 72 (1960), pp. 458-520.
19. P. HAJŁASZ, Sobolev inequalities, truncation method, and John domains, in Papers on Analysis, Rep. Univ. Jyväskylä Dep. Math. Stat. 83, Univ. Jyväskylä, Jyväskylä, Finland, 2001, pp. 109-126.
20. P. W. JONES, Quasiconforrnal mappings and extendability of functions in Sobolev spaces, Acta Math., 147 (1981), pp. 71-88.
21. A. KLAWONN AND O. RHEINBACH, A parallel implementation of dual-primal FETI methods for three-dimensional linear elasticity using a transformation of basis, SIAM J. Sci. Comput., 28 (2006), pp. 1886-1906.
22. A. KLAWONN AND O. RHEINIBACH, Inexact FETI-DP methods, Internat. J. Numer. Methods Engrg., 69 (2007), pp. 284-307.
23. A. KLAWONN AND O. RHEINBACH, Robust FETI-DP methods for heterogeneous three dimensional elasticity problems, Comput. Methods Appl. Mech. Engrg., 196 (2007), pp. 1400-1414.
24. A. KLAWONN AND O. B. WIDLUND, Dual-primal FETI methods for linear elasticity, Comm. Pure Appl. Math., 59 (2006), pp. 1523-1572.
25. A. KLAWONN, O. B. WIDLUND, AND M. DRYJA, Dual-primal FETI methods for three-dimensional elliptic problems with heterogeneous coefficients, SIAM J. Numer. Anal., 40 (2002), pp. 159-179.
26. J. LI AND O. B. WIDLUND, FETI-DP, BDDC, and block Cholesky methods, Internat. J. Numer. Methods Engrg., 66 (2006), pp. 250-271.
27. F. LIN AND X. YANG, Geometric Measure Theory - An Introduction, Advanced Mathematics, (Beijing/Boston), 1, Science Press, Beijing, 2002.
28. J. MANDEL, C. R. DOHRMANN, AND R. TEZAUR, An algebraic theory for primal and dual substructuring methods by constraints, Appl. Numer. Math., 54 (2005), pp. 167-193.
29. J. MANDEL AND R. TEZAUR, On the convergence of a dual-primal substructuring method, Numer. Math., 88 (2001), pp. 543-558.
30. V. G. MAZ'JA, Classes of domains and imbedding theorems for function spaces, Soviet Math. Dokl., 1 (1960), pp. 882-885.
31. V. G. MAZ'JA, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985 (translated from the Russian by T. O. Shaposhnikova).
32. O. RHEINUACH, Parallel Scalable Iterative Substructuring: Robust Exact and Inexact FETI-DP Methods with Applications to Elasticity, Ph.D. thesis, Department of Mathematics, University of Duisburg-Essen, Essen, Germany, 2006.
33. D. RIXEN, Substructuring and Dual Methods in Structural Analysis, Ph.D. thesis, University of Liège, Liège, Belgium, 1997.
34. D. RIXEN AND C. FARHAT, A simple and efficient extension of a class of substructure based preconditioners to heterogeneous structural mechanics problems, Internat. J. Numer. Methods Engrg., 44 (1999), pp. 489-516.
35. L. R. SCOTT AND S. ZHANG, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54 (1990), pp. 483-493.
36. A. TOSELLI AND O. B. WIDLUND, Domain decomposition methods - algorithms and theory, in Springer Series in Computational Mathematics, 34, Springer-Verlag, Berlin, 2005.
37. O. B. WIDLUND, An extension theorem for finite element spaces with three applications, pp. 110-122. Proceedings of the Second GAMM-Seminar on Numerical Techniques in Continuum Machanics, Kiel, Germany, 1986. W. Hackbusch and K. Witsch, eds., Friedr. Vieweg und Sohn, Braunschweig/Wiesbaden, 1987.