Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Buchtitelbild

2013 | OriginalPaper | Buchkapitel

1. Preliminaries

verfasst von : Clemens Pechstein

Erschienen in: Finite and Boundary Element Tearing and Interconnecting Solvers for Multiscale Problems

Verlag: Springer Berlin Heidelberg

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

This chapter contains some standard results that we need in subsequent chapters. The material is arranged such that an unexperienced reader can read through it linearly. The experienced reader my bypass the chapter in the beginning and return to it when necessary.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Nächstes Kapitel One-Level FETI/BETI Methods
Fußnoten
1
Note that in the literature of boundary integral equations (e.g., [McL00, Ste03b, Ste08]), the space \({H}_{00}^{1/2}(\widetilde{\Gamma })\) is often denoted by \(\widetilde{{H}}^{1/2}(\widetilde{\Gamma })\) and the two dual spaces by \({H}^{-1/2}(\widetilde{\Gamma }) := {(\widetilde{{H}}^{1/2}(\widetilde{\Gamma }))}^{{_\ast}}\) and \(\widetilde{{H}}^{-1/2}(\widetilde{\Gamma }) := {({H}^{1/2}(\widetilde{\Gamma }))}^{{_\ast}}\).
 
2
Often, the map F τ itself is a polynomial of degree \(\leq {p}_{\tau }\), which allows physical elements with curved boundaries.
 
Literatur
AB05.
G. Allaire and R. Brizzi. A multiscale finite element method for numerical homogenization. Multiscale Model. Simul., 4(3):790–812, 2005.
AB06.
T. Arbogast and K.J. Boyd. Subgrid upscaling and mixed multiscale finite elements. SIAM J. Numer. Anal., 44:1150–1171, 2006.
ABDP81.
R. E. Alcouffe, Achi Brandt, J. E. Dendy, Jr., and J. W. Painter. The multi-grid method for the diffusion equation with strongly discontinuous coefficients. SIAM J. Sci. Comput., 2(4):430–454, 1981.
AF03.
R. A. Adams and J. J. F. Fournier. Sobolev Spaces, volume 140 of Pure and Applied Mathematics. Academic Press, Amsterdam, Boston, second edition, 2003.
AG02.
M. Ainsworth and B. Guo. Analysis of iterative sub-structuring techniques for boundary element approximation of the hypersingular operator in three dimensions. Appl. Anal., 81(2):241–280, 2002.
AGKS08.
B. Aksoylu, I. G. Graham, H. Klie, and R. Scheichl. Towards a rigorously justified algebraic preconditioner for high-contrast diffusion problems. Comput. Visual Sci., 11(4-6):319–331, 2008.
AS97.
M. Ainsworth and B. Senior. Aspects of an adaptive hp-finite element method: adaptive strategy, conforming approximation and efficient solvers. Comput. Methods Appl. Mech. Engrg., 150(1-4):65–87, 1997.
Bas96.
P. Bastian. Parallele Adaptive Mehrgitterverfahren. Teubner Skripten zur Numerik. B. G. Teubner, Stuttgart, 1996.
BB08.
S. Boerm and J. Bendoraityte. Distributed \({\mathcal{H}}^{2}\)-matrices for non-local operators. Comput. Visual Sci., 11(4-6):237–249, 2008.
BBC+06.
Y. Bazilevs, L. Beirão da Veiga, J. A. Cottrell, T. J. R. Hughes, and G. Sangalli. Isogeometric analysis: approximation, stability and error estimates for h-refined meshes. Math. Models Methods Appl. Sci., 16(7):1031–1090, 2006.
BBK+09.
D. Balzani, D. Brands, A. Klawonn, O. Rheinbach, and J. Schröder. On the mechanical modeling of anisotropic biological soft tissue and iterative parallel solution strategies. Archive of Applied Mechanics, 80(5):479–488, 2009.
BCLP10.
L. Beirão da Veiga, C. Chinosi, C. Lovadina, and L. F. Pavarino. Robust BDDC preconditioners for Reissner - Mindlin plate bending problems and MITC elements. SIAM J. Numer. Anal., 47(6):4214–4238, 2010.
BCPS12.
L. Beirão da Veiga, D. Cho, L. F. Pavarino, and S. Scacchi. BDDC preconditioners for isogeometric analysis. Math. Models Methods Appl. Sci., 2012. accepted for publication.
BDF+00.
M. Bhardwaj, D. Day, C. Farhat, M. Lesoinne, K. Pierson, and D. Rixen. Application of the FETI method to ASCI problems—scalability results on 1000 processors and discussion of highly heterogeneous problems. Int J. Numer. Meth. Engng., 47:513–535, 2000.
BDS08.
J. Bouchala, Z. Dostál, and M. Sadowská. Theoretically supported scalable BETI method for variational inequalities. Computing, 82(1):53–75, 2008.
BDV97.
P. E. Bjørstad, M. Dryja, and E. Vainikko. Additive schwarz methods without subdomain overlap and with new coarse spaces. In R. Glowinski, Z.-C. Shi J. Périaux, and O. Widlund, editors, Domain Decomposition Methods in Sciences and Engineering (Beijing, 1995), pages 141–157. John Wiley & Sons, Chichester, 1997. http://​www.​ddm.​org/​DD08/​.
Beb00.
M. Bebendorf. Effiziente numerische Lösung von Randintegralgleichungen unter Verwendung von Niedrigrang-Matrizen. PhD thesis, Universität Saarbrücken, 2000.
Beb03.
M. Bebendorf. A note on the Poincaré inequality for convex domains. Z. Anal. Anwendungen, 22(4):751–756, 2003.
Beb05.
M. Bebendorf. Hierarchical LU decomposition based preconditioners for BEM. Computing, 74(3):225–247, 2005.
Beb08.
M. Bebendorf. Hierarchical Matrices – A Means to Efficiently Solve Elliptic Boundary Value Problems, volume 63 of Lecture Notes in Computational Science and Engineering. Springer-Verlag, Berlin, 2008.
BEL08.
S. Beuchler, T. Eibner, and U. Langer. Primal and dual interface concentrated iterative substructuring methods. SIAM J. Numer. Anal., 46(6):2818–2842, 2008.
BF91.
F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics. Springer-Verlag, New York, 1991.
BG86.
I. Babuska and B. Q. Guo. The h-p version of the finite element method, part 1: the basic approximation results. Comp. Mech., 1:22–41, 1986.
BGH06.
BGLV89.
J.-F. Bourgat, R. Glowinski, P. Le Tallec, and M. Vidrascu. Variational formulation and algorithm for trace operator in domain decomposition calculations. In T. Chan, R. Glowinski, J. Périaux, and O. Widlund, editors, Domain Decomposition Methods. Second International Symposium on Domain Decomposition Methods, pages 3–16. SIAM, Philadelphia, PA, 1989. http://​www.​ddm.​org/​DD02/​.
BH70.
J. H. Bramble and S. R. Hilbert. Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM J. Numer. Anal., 7:112–124, 1970.
BH03.
S. C. Brenner and Q. He. Lower bounds for three-dimensional nonoverlapping domain decomposition algorithms. Numer. Math., 93(3):445–470, 2003.
BHK08.
C. Bertoglio, W. Hackbusch, and B. N. Khoromskij. Low rank tensor-product approximation of projected green kernels via sinc-quadratures. Preprint 79/2008, MPI MIS, Leipzig, 2008.
BJ79.
F. Brezzi and C. Johnson. On the coupling of boundary integral and finite element methods. Calcolo, 16:189–201, 1979.
BJN78.
F. Brezzi, C. Johnson, and J. C. Nédélec. On the coupling of boundary integral and finite element methods. In Proceedings of the Fourth Symposium on Basic Problems of Numerical Mathematics (Pilzen, 1978), pages 103–114. Charles University, Prague, 1978.
BK95.
S. Buckley and P. Koskela. Sobolev-Poincaré implies John. Math. Res. Lett., 2(5):577–593, 1995.
BK05.
M. Bebendorf and R. Kriemann. Fast parallel solution of boundary integral equations and related problems. Comput. Visual Sci., 8(3-4):121–135, 2005.
BKRS08.
D. Brands, A. Klawonn, O. Rheinbach, and J. Schrd̈er. Modeling and convergence in arterial wall simulations using a parallel FETI solution strategy. Computer Methods in Biomechanics and Biomedical Engineering, 11(5):569–583, 2008.
BM97.
C. Bernardi and Y. Maday. Spectral methods. In P. G. Ciarlet and J.-L. Lions, editors, Handbook of Numerical Analysis, Volume V: Techniques of Scientific Computing (Part 2). North-Holland, 1997.
BMP94.
C. Bernardi, Y. Maday, and A. T. Patera. A nonconforming approach to domain decomposition: The mortar element method. In H. Brezis and J.-L. Lions, editors, College de France Seminar, pages 13–51. Longman Scientific and Technical Harlow, UK, 1994.
Boj88.
B. Bojarski. Remarks on Sobolev imbedding inequalities. In Complex analysis, Joensuu 1987, volume 1351 of Lecture Notes in Math., pages 52–68. Springer-Verlag, Berlin, 1988.
BP88.
J. H. Bramble and J. E. Pasciak. A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems. Mathematics of Computation, 50(181):1–17, 1988.
BPS86.
J. H. Bramble, J. E. Pasciak, and A. H. Schatz. The construction of preconditioners for elliptic problems by substructuring, I. Math. Comp., 47(175):103–134, 1986.
BPS87.
J. H. Bramble, J. E. Pasciak, and A. H. Schatz. The construction of preconditioners for elliptic problems by substructuring, II. Math. Comp., 49(179):1–16, 1987.
BPS88.
J. H. Bramble, J. E. Pasciak, and A. H. Schatz. The construction of preconditioners for elliptic problems by substructuring, III. Math. Comp., 51(184):415–430, 1988.
BPS89.
J. H. Bramble, J. E. Pasciak, and A. H. Schatz. The construction of preconditioners for elliptic problems by substructuring, IV. Math. Comp., 53(187):1–24, 1989.
Bra01.
D. Braess. Finite elements. Theory, fast solvers, and applications in solid mechanics. Cambridge University Press, Cambridge, 2001.
Bre00.
F. Brezzi. Interacting with the subgrid world. In D.F. Griffiths and G.A. Watson, editors, Numerical Analysis 1999, pages 69–82. Chapman & Hall/CRC, Boca Raton, 2000.
Bre02.
S. C. Brenner. A new look at FETI. In N. Debit, M. Garbey, R. Hoppe, J. Périaux, D. Keyes, and Y. Kuznetsov, editors, Domain decomposition methods in science and engineering (Lyon, 2000), Theory Eng. Appl. Comput. Methods, pages 41–51. Internat. Center Numer. Methods Eng. (CIMNE), Barcelona, 2002. http://​www.​ddm.​org/​DD13/​.
Bre03a.
S. C. Brenner. An additive Schwarz preconditioner for the FETI method. Numerische Mathematik, 94(1):1–31, 2003.
Bre03b.
S. C. Brenner. Analysis of two-dimensional FETI-DP preconditioners by the standard additive Schwarz framework. Electron. Trans. Numer. Anal., 16:165–185, 2003.
BS87.
I. Babuška and M. Suri. The h-p version of the finite element method with quasi-uniform meshes. RAIRO Modél. Math. Anal. Numér., 21(2):199–238, 1987.
BS00.
S. C. Brenner and L.-Y. Sung. Discrete Sobolev and Poincaré inequalities via Fourier series. East-West J. Numer. Math., 8(2):83–92, 2000.
BS02.
S. C. Brenner and L. R. Scott. The mathematical theory of finite element methods, volume 15 of Texts in Applied Mathematics. Springer-Verlag, New York, second edition, 2002.
BS07.
S. C. Brenner and L. Sung. BDDC and FETI-DP without matrices or vectors. Comput. Methods Appl. Mech. Engrg., 8:1429–1435, 2007.
BSEH11.
M. J. Borden, M. A. Scott, J. A. Evans, and T. J. R. Hughes. Isogeometric finite element data structures based on Bézier extraction of NURBS. Internat. J. Numer. Methods Engrg., 87(1-5):15–47, 2011.
BX91.
J. H. Bramble and J. Xu. Some estimates for a weighted L 2 projection. Math. Comp., 56(194):463–476, 1991.
BZ00.
J. H. Bramble and X. Zhang. The analysis of multigrid methods. In P. G. Ciarlet and J. Lions, editors, Handbook of Numerical Analysis, volume VII, pages 173–415. North-Holland, Amsterdam, 2000.
CCC+03.
W. C. Chew, H. Y. Chao, T. J. Cui, C. C. Lu, S. Ohnuki, Y. C. Pan, J. M. Song, S. Velamparambil, and J. S. Zhao. Fast integral equation solvers in computational electromagnetics of complex structures. Engineering Analysis with Boundary Elements, 27(8):803–823, 2003.
CDNQ12.
T. Cluzeau, V. Dolean, F. Nataf, and A. Quadrat. Symbolic preconditioning techniques for linear systems of partial differential equations. In R. Bank, R. Kornhuber, and O. Widlund, editors, Domain Decomposition Methods in Science and Engineering XX, Lecture Notes in Computational Science and Engineering. Springer-Verlag, 2012. to appear; http://​hal.​archives-ouvertes.​fr/​hal-00664092.
CDS03.
X.-C. Cai, M. Dryja, and M. Sarkis. Restricted additive Schwarz preconditioners with harmonic overlap for symmetric positive definite linear systems. SIAM J. Numer. Anal., 41(4):1209–1231, 2003.
CFH+03.
T. Chartier, R. D. Falgout, V. E. Henson, J. Jones, T. Manteuffel, S. McCormick, J. Ruge, and P. S. Vassilevski. Spectral amge (ρamge). SIAM J. Sci. Comput, 25(1):1–26, 2003.
CGH10.
C.-C. Chu, I. G. Graham, and T.-Y. Hou. A new multiscale finite element method for high-contrast elliptic interface problems. Math. Comp., 79:1915–1955, 2010.
CGR99.
H. Cheng, L. Greengard, and V. Rokhlin. A fast adaptive multipole algorithm in three dimensions. J. Comput. Phys., 155(2):468–498, 1999.
CGSS00.
K. A. Cliffe, I. G. Graham, R. Scheichl, and L. Stals. Parallel computation of flow in heterogeneous media modelled by mixed finite elements. J. Comput. Phys., 164(2):258–282, 2000.
CH02.
Z. Chen and T.Y. Hou. A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comp., 72:541–576, 2002.
CHB09.
J. A. Cottrell, T. J. R. Hughes, and Y. Bazilevs. Isogeometric Analysis: Towards Integration of CAD and FEA. John Wiley & Sons, 2009.
Chu93.
S.-K. Chua. Weighted Sobolev inequalities on domains satisfying the chain condition. Proc. Amer. Math. Soc., 117(2):449–457, 1993.
Cia87.
P. G. Ciarlet. The finite element method for elliptic problems, volume 4 of Studies in Mathematics and its Applications. North-Holland, Amsterdam, 1987.
CKL98.
C. Carstensen, M. Kuhn, and U. Langer. Fast parallel solvers for symmetric boundary element domain decomposition equations. Numer. Math., 79:321–347, 1998.
Clé75.
P. Clément. Approximation by finite element functions using local regularization. RAIRO Analyse Numérique, 9(R-2):77–84, 1975.
CM94.
T. F. Chan and T. P. Mathew. Domain decomposition algorithms. Acta Numerica, pages 61–143, 1994.
Cos87.
M. Costabel. Symmetric methods for the coupling of finite elements and boundary elements. In C. A. Brebbia, W. L. Wendland, and G. Kuhn, editors, Boundary Elements IX, pages 411–420. Springer-Verlag, Berlin, 1987.
Cos88.
M. Costabel. Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal., 19(3):613–626, 1988.
Cos07.
M. Costabel. Some historical remarks on the positivity of boundary integral operators. In M. Schanz and O. Steinbach, editors, Boundary Element Analysis – Mathematical Aspects and Applications, volume 29 of Lecture Notes in Applied and Computational Mechanics, pages 1–27. Springer-Verlag, Berlin, 2007.
Cro03.
J. M. Cros. A preconditioner for the Schur complement domain decomposition method. In I. Herrera, D. E. Keyes, and O. B. Widlund, editors, Domain Decomposition Methods in Science and Engineering. National Autonomous University of Mexico (UNAM), México, 2003. http://​www.​ddm.​org/​DD14/​.
CS90.
M. Costabel and E. P. Stephan. Coupling of finite and boundary element methods for an elastoplastic interface problem. SIAM J. Numer. Anal., 27(5):1212–1226, 1990.
CW06.
S.-K. Chua and R. L. Wheeden. Estimates of best constants for weighted Poincaré inequalities on convex domains. Proc. London Math. Soc. (3), 93(1):197–226, 2006.
Dah77.
B. E. J. Dahlberg. Estimates of harmonic measure. Arch. Rational Mech. Anal., 65(3):275–288, 1977.
Dau88.
M. Dauge. Elliptic boundary value problems on corner domains. Springer-Verlag, New York, 1988.
De 91.
Y.-H. De Roeck. Résolution sur Ordinateurs Multi-Processeurs de Problème d’Elasticité par Décomposition des Domaines. PhD thesis, Université Paris IX Dauphine, 1991.
DE12.
D. A. Di Pietro and A. Ern. Mathematical Aspects of Discontinuous Galerkin Methods, volume 69 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer, Heidelberg, 2012.
Dem07.
L. Demkowicz. Computing with hp-adaptive finite elements. Vol. 1. Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series. Chapman & Hall/CRC, Boca Raton, FL, 2007.
DGS07.
M. Dryja, J. Galvis, and M. Sarkis. BDDC methods for discontinuous Galerkin discretization of elliptic problems. J. Complexity, 23(4-6):715–739, 2007.
DGS08.
M. Dryja, J. Galvis, and M. Sarkis. Balancing domain decomposition methods for discontinuous Galerkin discretization. In Domain decomposition methods in science and engineering XVII, volume 60 of Lecture Notes in Computational Science and Engineering, pages 271–278. Springer, Berlin, 2008.
DGS11.
M. Dryja, J. Galvis, and M. Sarkis. N-N solvers for a DG discretization for geometrically nonconforming substructures and discontinuous coefficients. In Y. Huang, R. Kornhuber, O. Widlund, and J. Xu, editors, Domain Decomposition Methods in Science and Engineering XIX, volume 78 of Lecture Notes in Computational Science and Engineering, pages 27–38. Springer-Verlag, Berlin, 2011.
DGS12.
M. Dryja, J. Galvis, and M. Sarkis. Neumann-Neumann methods for a DG discretization of elliptic problems with discontinuous coefficients on geometrically nonconforming substructures. Numerical Methods for Partial Differential Equations, 28(4):1194–1226, 2012.
DHK+05.
Z. Dostál, D. Horák, R. Kučera, V. Vondrák, J. Haslinger, J. Dobiáš, and S. Pták. FETI based algorithms for contact problems: scalability, large displacements and 3D Coulomb friction. Comput. Methods Appl. Mech. Engrg., 194(2–5):395–409, 2005.
DHK06.
Z. Dostál, D. Horák, and R. Kučera. Total FETI – An easier implementable variant of the FETI method for numerical solution of elliptic PDE. Commun. Numer. Methods Eng., 22(12):1155–1162, 2006.
DHL03.
C. C. Douglas, G. Haase, and U. Langer. A Tutorial on Elliptic PDE Solvers and Their Parallelization. SIAM, Philadelphia, 2003.
DKV+10.
Z. Dostál, T. Kozubek, V. Vondrák, T. Brzobohatý, and A. Markopoulos. Scalable TFETI algorithm for the solution of multibody contact problems of elasticity. Internat. J. Numer. Methods Engrg., 82(11):1384–1405, 2010.
DKW08a.
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(4):2153–2168, 2008.
DKW08b.
C. R. Dohrmann, A. Klawonn, and O. B. Widlund. Extending theory for domain decomposition algorithms to irregular subdomains. In U. Langer, M. Discacciati, O. Widlund, and W. Zulehner, editors, Domain Decomposition Methods in Science and Engineering XVII, volume 60 of Lecture Notes in Computational Engineering and Science, pages 255–261. Springer-Verlag, Berlin, 2008.
DL91.
Y.-H. De Roeck and P. Le Tallec. Analysis and test of a local domain decomposition preconditioner. In R. Glowinski, Y. A. Kuznetsov, G. A. Meurant, J. Périaux, and O. Widlund, editors, Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, pages 112–128. SIAM, Philadelphia, PA, 1991. http://​www.​ddm.​org/​DD04/​.
DNR08.
V. Dolean, F. Nataf, and G. Rapin. How to use the Smith factorization for domain decomposition methods applied to the Stokes equation. In Domain decomposition methods in science and engineering XVII, volume 60 of Lecture Notes in Computational Science and Engineering, pages 477–848. Springer, Berlin, 2008.
DNR09.
V. Dolean, F. Nataf, and G. Rapin. Deriving a new domain decomposition method for the Stokes equation using the Smith factorization. Math. Comp., 78:789–814, 2009.
DNSS11.
V. Dolean, F. Nataf, R. Scheichl, and N. Spillane. Analysis of a two-level schwarz method with coarse spaces based on local Dirichlet-to-Neumann maps. Preprint HAL-00586246, Hyper Articles en Ligne, 2011. submitted.
Doh03.
C. R. Dohrmann. A preconditioner for substructuring based on constrained energy minimization. SIAM J. Sci. Comput., 25(1):246–258, 2003.
Doh07.
C. R. Dohrmann. An approximate BDDC preconditioner. Numer. Linear Algebra Appl., 14(2):149–168, 2007.
DORH89.
L. Demkowicz, J. T. Oden, W. Rachowicz, and O. Hardy. Toward a universal h-p adaptive finite element strategy. I. Constrained approximation and data structure. Comput. Methods Appl. Mech. Engrg., 77(1-2):79–112, 1989.
Dry87.
M. Dryja. A method of domain decomposition for 3-D finite element problems. In R. Glowinski, G. H. Golub, G. A. Meurant, and J. Périaux, editors, First International Symposium on Domain Decomposition methods for Partial Differential Equations, pages 43–61. SIAM, Philadelphia, PA, 1987. http://​www.​ddm.​org/​DD01/​.
DS80.
T. Dupont and R. Scott. Polynomial approximation of functions in Sobolev spaces. Math. Comp., 34(150):441–463, 1980.
DS11.
M. Dryja and M. Sarkis. Technical tools for boundary layers and applications to heterogeneous coefficients. In Y. Huang, R. Kornhuber, O. Widlund, and J. Xu, editors, Decomposition Methods in Science and Engineering XIX, volume 78 of Lecture Notes in Computational Science and Engineering, pages 205–212. Springer-Verlag, Berlin, 2011.
DSW94.
M. Dryja, B. F. Smith, and O. B. Widlund. Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions. SIAM J. Numer. Anal., 31(6):1662–1694, 1994.
DSW96.
M. Dryja, M. V. Sarkis, and O. B. Widlund. Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions. Numer. Math., 72:313–348, 1996.
DW94.
M. Dryja and O. B. Widlund. Domain decomposition algorithms with small overlap. SIAM J. Sci. Comput., 15(3):604–620, 1994.
DW95.
M. Dryja and O. B. Widlund. Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Comm. Pure Appl. Math., 48(2):121–155, 1995.
DW06.
L. T. Dechevski and W. L. Wendland. On the Bramble-Hilbert lemma. II. An improved estimate of the embedding constant. Int. J. Pure Appl. Math., 33(4):433–464, 2006.
DW09.
C. R. Dohrmann and O. B. Widlund. An overlapping schwarz algorithm for almost incompressible elasticity. SIAM J. Numer. Anal., 47(4):2897–2923, 2009.
DW10.
C. R. Dohrmann and O. B. Widlund. Hybrid domain decomposition algorithms for compressible and almost incompressible elasticity. Internat. J. Numer. Methods Engrg., 82:157–183, 2010.
DW12a.
C. R. Dohrmann and O. B. Widlund. An iterative substructuring algorithm for two-dimensional problems in H(curl). SIAM J. Numer. Anal., 50(3):1004–1028, 2012.
DW12b.
EGLW12.
Y. Efendiev, J. Galvis, R. Lazarov, and J. Willems. Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms. ESAIM Math. Model. Numer. Anal., 46(5):1175–1199, 2012.
EH09.
Y. Efendiev and T.Y. Hou. Multiscale Finite Element Methods: Theory and Applications. Springer-Verlag, New York, 2009.
EHN96.
H. W. Engl, M. Hanke, and A. Neubauer. Regularization of inverse problems, volume 375 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1996.
Eib06.
T. Eibner. Adaptive und randkonzentrierte FEM. PhD thesis, TU Chemnitz, Chemnitz, Germany, June 2006.
EM06.
T. Eibner and J. M. Melenk. A local error analysis of the boundary concentrated FEM. IMA J. Numer. Anal., 26(4):752–778, 2006.
EM07a.
T. Eibner and J. M. Melenk. An adaptive strategy for hp-FEM based on testing for analyticity. Comput. Meth., 39(5):575–595, 2007.
EM07b.
T. Eibner and J. M. Melenk. Multilevel preconditioning for the boundary concentrated hp-FEM. Comp. Methods Appl. Mech. Engrg., 196(37-40):3713–3725, 2007.
Eva98.
L. C. Evans. Partial Differential Equations, volume 19 of Graduate Studies in Mathematics. AMS, Providence, RI, 1998.
FATL05.
C. Farhat, P. Avery, R. Tezaur, and J. Li. FETI-DPH: a dual-primal domain decomposition method for acoustic scattering. J. Comput. Acoust., 13(3):499–524, 2005.
FCM95.
C. Farhat, P. Chen, and J. Mandel. A scalable Lagrange multiplier based domain decomposition method for time-dependent problems. Int. J. Numer. Meth. Engng., 38(22):3831–3853, 1995.
FCMR98.
C. Farhat, P. Chen, J. Mandel, and F.-X. Roux. The two-level FETI method part II: Extensions to shell problems, parallel implementation and performance results. Comput. Methods Appl. Mech. Engrg., 155(1–2):153–179, 1998.
FCR94.
C. Farhat, L. Crivelli, and F.-X. Roux. Extending substructure based iterative solvers to multiple load and repeated analyses. Comput. Meth. Appl. Mech. Engng., 117:195–200, 1994.
FCRR98.
C. Farhat, P.-S. Chen, F. Risler, and F.-X. Roux. A unified framework for accelerating the convergence of iterative substructuring methods with Lagrange multipliers. Int. J. Numer. Meth. Engng., 42(2):257–288, 1998.
FF60.
H. Federer and W. H. Fleming. Normal and integral currents. Ann. of Math., 2:482–520, 1960.
Fie73.
M. Fiedler. Algebraic connectivity of graphs. Czechoslovak Math. J., 23(98):298–305, 1973.
FKS82.
E. B. Fabes, C. E. Kenig, and R. P. Serapioni. The local regulartity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations, 7(1):77–116, 1982.
FLL+01.
C. Farhat, M. Lesoinne, P. Le Tallec, K. Pierson, and D. Rixen. FETI-DP: A dual-primal unified FETI method I: A faster alternative to the two-level FETI method. Internat. J. Numer. Methods Engrg., 50(7):1523–1544, 2001.
FLLA05.
C. Farhat, J. Li, M. Lesoinne, and P. Avery. A FETI method for a class of indefinite or complex second- or fourth-order problems. In R. Kornhuber, R. H. W. Hoppe, D. E. Keyes, J. Periaux, O. Pironneau, and J. Xu, editors, Domain Decomposition Methods in Science and Engineering, volume 40 of Lecture Notes in Computational Science and Engineering, pages 19–33. Springer-Verlag, Berlin, 2005. http://​www.​ddm.​org/​DD15/​.
FLP00.
C. Farhat, M. Lesoinne, and K. Pierson. A scalable dual-primal domain decomposition method. Numer. Linear Algebra Appl., 7(7–8):687–714, 2000.
FM97.
C. Farhat and J. Mandel. Scalable substructuring by lagrange multipliers in theory and practice. In P. Bjørstad, M. Espedal, and D. Keyes, editors, Ninth International Conference: Domain Decomposition Methods in Science and Engineering, 1997. http://​www.​ddm.​org/​DD9/​Farhat.​pdf.
FM98.
C. Farhat and J. Mandel. The two-level FETI method for static and dynamic plate problems part I: An optimal iterative solver for biharmonic systems. Comput. Methods Appl. Mech. Engrg., 155(1–2):129–151, 1998.
FML00a.
C. Farhat, A. Macedo, and M. Lesoinne. A two-level domain decomposition method for the iterative solution of high-frequency exterior Helmholtz problems. Numer. Math., 85(2):283–303, 2000.
FML+00b.
C. Farhat, A. Macedo, M. Lesoinne, F.-X. Roux, F. Magoulès, and A. de La Bourdonnaye. Two-level domain decomposition methods with Lagrange multipliers for the fast iterative solution of acoustic scattering problems. Comput. Methods Appl. Mech. Engrg., 184:213–240, 2000.
FMR94.
C. Farhat, J. Mandel, and F.-X. Roux. Optimal convergence properties of the FETI domain decomposition method. Comput. Methods Appl. Mech. Engrg., 115:365–385, 1994.
FMT99.
C. Farhat, A. Macedo, and R. Tezaur. FETI-H: A scalable domain decomposition method for high frequency exterior Helmholtz problems. In C.-H. Lai, P. E. Bjørstad, M. Cross, and O. B. Widlund, editors, Domain Decomposition Methods in Science and Engineering: Eleventh International Conference London, UK, pages 231–241, 1999. http://​www.​ddm.​org/​DD11/​Farhat.​pdf.
FP03.
Y. Fragakis and M. Papadrakakis. The mosaic of high performance domain decomposition methods for structural methanics: Formulation, interrelation and numerical efficiency of primal and dual methods. Comput. Methods Appl. Mech. Engrg., 192:3799–3830, 2003.
FP04.
Y. Fragakis and M. Papadrakakis. The mosaic of high performance domain decomposition methods for structural methanics – Part II: Formulation enhancements, multiple right-hand sides and implicit dynamics. Comput. Methods Appl. Mech. Engrg., 193:4611–4662, 2004.
FR91.
C. Farhat and F.-X. Roux. A method of finite element tearing and interconnecting and its parallel solution algorithm. Int. J. Numer. Meth. Engrg., 32(6):1205–1227, 1991.
FR92.
C. Farhat and F.-X. Roux. An unconventional domain decomposition method for an efficient parallel solution of large-scale finite element systems. SIAM J. Sci. and Stat. Comput., 13(1):379–396, 1992.
FR94.
C. Farhat and F.-X. Roux. Implicit parallel processing in structural mechanics. In J. T. Oden, editor, Computational Mechanics Advances, volume 2, pages 1–124. North-Holland, Amsterdam, 1994.
FS97.
S. A. Funken and E. P. Stephan. The BPX preconditioner for the single layer potential operator. Appl. Anal., 67(3–4):327–340, 1997.
GE10a.
J. Galvis and Y. Efendiev. Domain decomposition preconditioners for multiscale flows in high contrast media. Multiscale Model. Simul., 8(4):1461–1483, 2010.
GE10b.
J. Galvis and Y. Efendiev. Domain decomposition preconditioners for multiscale flows in high contrast media: reduced dimension coarse spaces. Multiscale Model. Simul., 8(5):1621–1644, 2010.
GH99.
I. G. Graham and M. J. Hagger. Unstructured additive Schwarz-conjugate gradient method for elliptic problems with highly discontinuous coefficients. SIAM J. Sci. Comput., 20(6):2041–2066, 1999.
GHLS12.
I. G. Graham, T. Y. Hou, O. Lakkis, and R. Scheichl (eds.). Numerical analysis of multiscale problems. Selected papers based on the presentations at the 91st London Mathematical Society symposium, Durham, UK, July 5–15, 2010, volume 83 of Lecture Notes in Computational Science and Engineering. Springer, Berlin, 2012.
GKM08.
GKR12.
GL81.
A. George and J. W. H. Liu. Computer solution of large sparse positive definite systems. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1981.
GLS07.
I. G. Graham, P. O. Lechner, and R. Scheichl. Domain decomposition for multiscale PDEs. Numer. Math., 106(4):589–626, 2007.
GR86.
V. Girault and P. A. Raviart. Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, New York, 1986.
Gri85.
P. Grisvard. Elliptic Problems on Nonsmooth Domains. Pitman, Boston, London, Melbourne, 1985.
Gri92.
P. Grisvard. Singularities in boundary value problems, volume 22 of RMA. Masson, Paris, 1992.
GS07.
I. G. Graham and R. Scheichl. Robust domain decomposition algorithms for multiscale PDEs. Numerical Methods for Partial Differential Equations, 23:859–878, 2007.
GV96.
G. H. Golub and C. F. Van Loan. Matrix Computations. Johns Hopkins University Press, Baltimore, MD, third edition, 1996.
GW88.
R. Glowinski and M. F. Wheeler. Domain decomposition and mixed finite element methods for elliptic problems. In R. Glowinski, G. H. Golub, G. A. Meurant, and J. Périaux, editors, First International Symposium on Domain Decomposition Methods for Partial Differential Equations, pages 144–172. SIAM, Philadelphia, PA, 1988. http://​www.​ddm.​org/​DD01/​.
Haa99.
G. Haase. Parallelisierung numerischer Algorithmen für partielle Differentialgleichungen. Teubner, Stuttgart, 1999.
Hac99.
W. Hackbusch. A sparse matrix arithmetic based on \(\mathcal{H}\)-matrices. Computing, 62(2):89–108, 1999.
Hac03.
W. Hackbusch. Multi-Grid Methods and Applications. Springer-Verlag, Berlin, second edition, 2003.
Haj01.
P. Hajłasz. Sobolev inequalities, truncation method, and John domains. In Papers on Analysis, pages 109–126. Report Univ. Jyväkylä, Dep. Math. Stat. 83, Jyväkylä, 2001.
HCB05.
T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Engrg., 194(39-41):4135–4195, 2005.
Heu01.
N. Heuer. Additive Schwarz method for the p-version of the boundary element method for the single layer potential operator on a plane screen. Numer. Math., 88(3):485–511, 2001.
HFMQ98.
T. J. R. Hughes, G. R. Feijóo, L. Mazzei, and J.-B. Quincy. The variational multiscale method – a paradigm for computational mechanics. Comput. Meth. Appl. Mech. Engrg., 166:3–24, 1998.
HHKL97.
G. Haase, B. Heise, M. Kuhn, and U. Langer. Adaptive domain decomposition methods for finite elements and boundary elements. In W. L. Wendland, editor, Boundary Element Topics, pages 121–147. Springer-Verlag, Berlin, 1997.
Hil87.
E. Hille. Analytic Function Theory, volume II. Chelsea Publishing Company, New York, second edition, 1987.
Hip02.
R. Hiptmair. Symmetric coupling for eddy current problems. SIAM J. Numer. Anal., 40:41–65, 2002.
HK00.
W. Hackbusch and B. N. Khoromskij. A sparse \(\mathcal{H}\)-matrix arithmetic: II. Approximation to multi-dimensional problems. Computing, 64(1):21–47, 2000.
HKD04.
J. Haslinger, R. Kučera, and Z. Dostál. An algorithm for the numerical realization of 3D contact problems with Coulomb friction. J. Comput. Appl. Math., pages 164–165, 2004.
HL92.
G. Haase and U. Langer. The non-overlapping domain decomposition multiplicative Schwarz method. International Journal of Computer Mathematics, 44:223–242, 1992.
HLM91a.
G. Haase, U. Langer, and A. Meyer. The approximate Dirichlet decomposition method. part I: An algebraic approach. Computing, 47:137–151, 1991.
HLM91b.
G. Haase, U. Langer, and A. Meyer. The approximate Dirichlet decomposition method. part II: Application to 2nd-order elliptic B.V.P.s. Computing, 47:153–167, 1991.
HLP10.
C. Hofreither, U. Langer, and C. Pechstein. Analysis of a non-standard finite element method based on boundary integral operators. Electron. Trans. Numer. Anal., 37:413–436, 2010.
HN89.
W. Hackbusch and Z. P. Nowak. On the fast matrix multiplication in the boundary element method by panel clustering. Numer. Math., 54(4):463–491, 1989.
Hof11.
C. Hofreither. l 2 error estimates for a nonstandard finite element method on polyhedral meshes. J. Numer. Math., 19(1):27–39, 2011.
HS98.
N. Heuer and E. P. Stephan. Iterative substructuring for hypersingular integral equations in \({\mathbf{R}}^{3}\). SIAM J. Sci. Comput., 20(2):739–749, 1998.
HS01.
N. Heuer and E. P. Stephan. An additive Schwarz method for the h-p version of the boundary element method for hypersingular integral equations in \({\mathbf{R}}^{3}\). IMA J. Numer. Anal., 21(1):265–283, 2001.
HS05.
Paul Houston and Endre Süli. A note on the design of hp-adaptive finite element methods for elliptic partial differential equations. Comput. Methods Appl. Mech. Engrg., 194(2-5):229–243, 2005.
HS07.
T. J. R Hughes and G. Sangalli. Variational multiscale analysis: the fine-scale Green’s function, projection, optimization, localization, stabilized methods. SIAM J. Num. Anal., 45:539–557, 2007.
HSW00.
G. C. Hsiao, O. Steinbach, and W. L. Wendland. Domain decomposition methods via boundary integral equations. J. Comput. Appl. Math., pages 521–537, 2000.
HW91.
G. C. Hsiao and W. L. Wendland. Domain decomposition in boundary element methods. In R. Glowinski, Y. A. Kuznetsov, G. Meurant, J. Périaux, and O. B. Widlund, editors, Proceedings of the Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, pages 41–49. SIAM, Philadelphia, PA, 1991. http://​www.​ddm.​org/​DD04/​.
HW08a.
J. S. Hesthaven and T. Warburton. Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, volume 54 of Texts in Applied Mathematics. Springer, New York, 2008.
HW08b.
G. C. Hsiao and W. L. Wendland. Boundary Integral Equations, volume 164 of Applied Mathematical Sciences. Springer-Verlag, Berlin, 2008.
IM95.
I. C. F. Ipsen and C. D. Meyer. The angle between complementary subspaces. Amer. Math. Monthly, 102(10):904–911, 1995.
IN85.
T. Iwaniec and C. A. Nolder. Hardy-Littlewood inequality for quasiregular mappings in certain domains in \({\mathbb{R}}^{n}\). Ann. Acad. Sci. Fenn. Ser. A I Math., 10:267–282, 1985.
Jar10.
JKR10.
JN80.
C. Johnson and J. C. Nédélec. On coupling of boundary integral and finite element methods. Math. Comp., 35(152):1063–1079, 1980.
Jon81.
P. W. Jones. Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math., 147:71–88, 1981.
Kam00.
C. Kamath. The FETI level 1 method: theory and implementation. Technical Report UCRLID-138075, Lawrence Livermore National Laboratory, CA, 2000.
KDW08.
H. H. Kim, M. Dryja, and O. B. Widlund. A BDDC method for mortar discretizations using a transformation of basis. SIAM J. Numer. Anal., 47(1):136–157, 2008.
Kho08a.
B. N. Khoromskij. Fast tensor approximation of multi-dimensional convolution with linear scaling. Preprint 36/2008, MPI MIS, Leipzig, 2008.
Kho08b.
B. N. Khoromskij. On tensor approximation of green iterations for kohn-sham equations. Comput. Visual Sci., 11(4–6), 2008.
Kim07.
H. H. Kim. A FETI-DP preconditioner for mortar methods in three dimensions. Electron. Trans. Numer. Anal., 26:103–120, 2007.
Kim08a.
H. H. Kim. A BDDC algorithm for mortar discretization of elasticity problems. SIAM J. Numer. Anal., 46(4):2090–2111, 2008.
Kim08b.
H. H. Kim. Domain decomposition algorithms for mortar discretizations. In Domain decomposition methods in science and engineering XVII, volume 60 of Lecture Notes in Computational Science and Engineering, pages 81–92. Springer, Berlin, 2008.
Kim08c.
H. H. Kim. A FETI-DP formulation of three dimensional elasticity problems with mortar discretization. SIAM J. Numer. Anal., 46(5):2346–2370, 2008.
KK98.
G. Karypis and V. Kumar. METIS: A Software Package for Partitioning Unstructured Graphs, Partitioning Meshes, and Computing Fill-Reducing Orderings of Sparse Matrices; Version 4.0. University of Minnesota, Department of Computer Science / Army HPC Research Center Minneapolis, MN 55455, USA, September 1998. http://​people.​scs.​fsu.​edu/​~burkardt/​pdf/​metis.​pdf.
KL04.
V. G. Korneev and U. Langer. Domain decomposition and preconditioning. In E. Stein, R. de Borst, and T. J. R. Hughes, editors, Encyclopedia of Computational Mechanics, volume 1, chapter 22. John Wiley & Sons, 2004.
KL05.
H. H. Kim and C.-O. Lee. A preconditioner for the FETI-DP formulation with mortar methods in two dimensions. SIAM J. Numer. Anal., 42(5):2159–2175, 2005.
KL10.
H. H. Kim and C.-O. Lee. A FETI-DP formulation for the three-dimensional Stokes problem without primal pressure unknowns. SIAM J. Sci. Comput., 32(6):3301–3322, 2010.
KLP10a.
H. H. Kim, C.-O. Lee, and E.-H. Park. A FETI-DP formulation for the Stokes problem without primal pressure components. SIAM J. Numer. Anal., 47(6):4142–4162, 2010.
KLP10b.
H. H. Kim, C.-O. Lee, and E.-H. Park. On the selection of primal unknowns for a FETI-DP formulation of the Stokes problem in two dimensions. Comput. Math. Appl., 60(12):3047–3057, 2010.
KM03.
B. N. Khoromskij and J. M. Melenk. Boundary concentrated finite element methods. SIAM J. Numer. Anal., 41(1):1–36, 2003.
KM07.
J. Kraus and S. Margenov. Multilevel methods for anisotropic elliptic problems. In Lectures on Advanced Computational Methods in Mechanics, volume 1 of Radon Series on Computational and Applied Mathematics. Walter de Gruyter, Berlin, New York, 2007.
KNRV11.
A. Klawonn, P. Neff, O. Rheinbach, and S. Vanis. FETI-DP domain decomposition methods for elasticity with structural changes: P-elasticity. ESAIM Math. Model. Numer. Anal., 45:563–602, 2011.
Kny01.
A. V. Knyazev. Toward the optimal preconditioned eigensolver: locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput., 23(2):517–541, 2001.
KPJT12.
KPR08.
A. Klawonn, L. Pavarino, and O. Rheinbach. Spectral element FETI-DP and BDDC preconditioners with multi-element subdomains. Comput. Meth. Appl. Mech. Engrg., 198(3–4):511–523, 2008.
KR06.
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(5):1886–1906, 2006.
KR07a.
A. Klawonn and O. Rheinbach. Inexact FETI-DP methods. Inter. J. Numer. Methods Engrg., 69:284–307, 2007.
KR07b.
A. Klawonn and O. Rheinbach. Robust FETI-DP methods for heterogeneous three dimensional elasticity problems. Comput. Methods Appl. Mech. Engrg., 196:1400–1414, 2007.
KR07c.
A. Klawonn and O. Rheinbach. Some computational results for robust FETI-DP methods applied to heterogeneous elasticity problems in 3D. In D. Keyes and O. Widlund, editors, Domain Decomposition Methods in Sciences and Engineering XVI, volume 55 of Lecture Notes in Computational Science and Engineering, pages 391–398. Springer-Verlag, Heidelberg, 2007.
KR10.
A. Klawonn and O. Rheinbach. Highly scalable parallel domain decomposition methods with an application to biomechanics. Z.Angew. Math. Mech., 90(1):5–32, 2010.
KR12.
A. Klawonn and O. Rheinbach. Deflation, projector preconditioning, and balancing in iterative substructuring methods: connections and new results. SIAM J. Sci. Comput., 34(1):A459–A484, 2012.
Kra11.
KRW07.
A. Klawonn, O. Rheinbach, and B. Wohlmuth. Dual-primal iterative substructuring for almost incompressible elasticity. In D. Keyes and O. Widlund, editors, Domain Decomposition Methods in Sciences and Engineering XVI, volume 55 of Lecture Notes in Computational Science and Engineering, pages 399–406. Springer-Verlag, Heidelberg, 2007.
KRW08.
A. Klawonn, O. Rheinbach, and O. B. Widlund. An analysis of a FETI-DP algorithm on irregular subdomains in the plane. SIAM J. Numer. Anal., 46(5):2484–2504, 2008.
KS99.
G. Karniadakis and S. Sherwin. Spectral/hp Element Methods for CFD. Oxford University Press, Oxford, 1999.
KS02.
M. Kuhn and O. Steinbach. Symmetric coupling of finite and boundary element for exterior magnetic field problems. Math. Methods Appl. Sci., 25:357–371, 2002.
KT09.
H. H. Kim and X. Tu. A three-level BDDC algorithm for mortar discretizations. SIAM J. Numer. Anal., 47(2):1576–1600, 2009.
KW00.
A. Klawonn and O. B. Widlund. A domain decomposition method with Lagrange multipliers and inexact solvers for linear elasticity. SIAM J. Sci. Comput., 22(4):1199–1219, 2000.
KW01.
A. Klawonn and O. B. Widlund. FETI and Neumann-Neumann iterative substructuring methods: Connections and new results. Comm. Pure Appl. Math., 54(1):57–90, 2001.
KW05.
A. Klawonn and O. B. Widlund. Selecting constraints in dual-primal FETI methods for elasticity in three dimensions. In R. Kornhuber, R. H. W. Hoppe, D. E. Keyes, J. Periaux, O. Pironneau, and J. Xu, editors, Domain Decomposition Methods in Science and Engineering, volume 40 of Lecture Notes in Computational Science and Engineering, pages 67–81. Springer-Verlag, Berlin, 2005. http://​www.​ddm.​org/​DD15/​.
KW06.
A. Klawonn and O. B. Widlund. Dual-primal FETI methods for linear elasticity. Comm. Pure Appl. Math., 59(11):1523–1572, 2006.
KWD02a.
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(1):159–179, 2002.
KWD02b.
A. Klawonn, O. B. Widlund, and M. Dryja. Dual-primal FETI methods with face constraints. In L. Pavarino and A. Toselli, editors, Recent Developments in Domain Decomposition Methods, volume 23 of Lecture Notes in Computational Science and Engineering, pages 27–40. Springer-Verlag, Berlin, 2002.
Lan94.
U. Langer. Parallel iterative solution of symmetric coupled FE/BE-equations via domain decomposition. Contemporary Mathematics, 157:335–344, 1994.
Le 94.
P. Le Tallec. Domain decomposition methods in computational mechanics. In J. T. Oden, editor, Computational Mechanics Advances, volume 1, pages 121–220. North-Holland, 1994.
Les03.
M. Lesoinne. A FETI-DP corner selection algorithm for three-dimensional problems. In I. Herrera, D. E. Keyes, O. Widlund, and R. Yates, editors, Proceedings of the 14th International Conference on Domain Decomposition Methods, pages 233–240. Mexico, 2003. http://​www.​ddm.​org/​DD14/​lesoinne.​pdf.
Li05.
J. Li. A dual-primal FETI method for incompressible stokes equations. Numer. Math., 102(2):257–275, 2005.
Lie06.
LM72.
J. L. Lions and E. Magenes. Non-Homogeneous Boundary Value Problems and Applications, volume I. Springer-Verlag, Berlin, 1972.
LOSZ07a.
U. Langer, G. Of, O. Steinbach, and W. Zulehner. Inexact data-sparse boundary element tearing and interconnecting methods. SIAM J. Sci. Comp., 29:290–314, 2007.
LOSZ07b.
U. Langer, G. Of, O. Steinbach, and W. Zulehner. Inexact fast multipole boundary element tearing and interconnecting methods. In D. Keyes and O. Widlund, editors, Domain Decomposition Methods in Sciences and Engineering XVI, volume 55 of Lecture Notes in Computational Science and Engineering, pages 405–412. Springer-Verlag, Heidelberg, 2007.
LP98.
M. Lesoinne and K. Pierson. An efficient FETI implementation on distributed and shared memory machines with independent number of subdomains and processors. In J. Mandel, C. Farhat, and X. Cai, editors, Domain Decomposition Methods 10. Proceedings of the 10th International Conference held in Boulder, CO, August 10–14, 1997, volume 218 of Contemporary Mathematics, pages 318–324. AMS, Providence, RI, 1998. http://​www.​ddm.​org/​DD10/​.
LP06.
U. Langer and C. Pechstein. Coupled finite and boundary element tearing and interconnecting solvers for nonlinear potential problems. ZAMM Z. Angew. Math. Mech., 86(12), 2006.
LP07.
U. Langer and C. Pechstein. Coupled FETI/BETI solvers for nonlinear potential problems in (un)bounded domains. In G. Ciuprina and D. Ioan, editors, Scientific Computing in Electrical Engineering, volume 11 of Mathematics in Industry: The European Consortium for Mathematics in Industry, pages 371–377. Springer-Verlag, Berlin, 2007.
LP08.
U. Langer and C. Pechstein. All-floating coupled data-sparse boundary and interface-concentrated finite element tearing and interconnecting methods. Computing and Visualization in Science, 11(4-6):307–317, 2008.
LPS05.
U. Langer, A. Pohoaţǎ, and O. Steinbach. Dual-primal boundary element tearing and interconnecting methods. Technical report no. 2005/6, Institute for Computational Mathematics, University of Technology, Graz, 2005.
LS03.
U. Langer and O. Steinbach. Boundary element tearing and interconnecting method. Computing, 71(3):205–228, 2003.
LS05.
U. Langer and O. Steinbach. Coupled boundary and finite element tearing and interconnecting methods. In R. Kornhuber, R. Hoppe, J. Periaux, O. Pironneau, O. Widlund, and J. Xu, editors, Domain Decomposition Methods in Science and Engineering, volume 40 of Lecture Notes in Computational Sciences and Engineering, pages 83–97. Springer-Verlag, Heidelberg, 2005. http://​www.​ddm.​org/​DD15/​.
Lun06.
Y. Lung-An. Numerical Methods for Exterior Problems, volume 2 of Peking University Series in Mathematics. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.
LW06a.
J. Li and O. Widlund. BDDC algorithms for incompressible Stokes equations. SIAM J. Numer. Anal., 44(6):2432–2455, 2006.
LW06b.
J. Li and O. B. Widlund. FETI-DP, BDDC, and block Cholesky methods. Internat. J. Numer. Methods Engrg., 66(2):250–271, 2006.
LW07.
J. Li and O. B. Widlund. On the use of inexact subdomain solvers for BDDC algorithms. Comput. Methods Appl. Mech. Engrg., 196(8):1415–1428, 2007.
Man93.
J. Mandel. Balancing domain decomposition. Comm. Numer. Methods Engrg., 9(3):233–241, 1993.
Mat08.
T. P. A. Mathew. Domain decomposition methods for the numerical solution of partial differential equations, volume 61 of Lecture Notes in Computational Science and Engineering. Springer-Verlag, Berlin, 2008.
Maz60.
V. G. Maz’ja. Classes of domains and imbedding theorems for functions spaces. Soviet Math. Dokl., 1:882–885, 1960.
Maz85.
V. G. Maz’ja. Sobolev spaces. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1985.
MB96.
J. Mandel and M. Brezina. Balancing domain decomposition for problems with large jumps in coefficients. Math. Comp., 65:1387–1401, 1996.
McL00.
W. McLean. Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge, UK, 2000.
MD03.
J. Mandel and C. R. Dohrmann. Convergence of a balancing domain decomposition by constraints and energy minimization. Numer. Lin. Alg. Appl., 10(7):639–659, 2003.
MDT05.
J. Mandel, C. R. Dohrmann, and R. Tezaur. An algebraic theory for primal and dual substructuring methods by constraints. Appl. Numer. Math., 54(2):167–193, 2005.
Mel02.
J. M. Melenk. hp-Finite Element Methods for Singular Pertubations, volume 1796 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2002.
Mon03.
P. Monk. Finite element methods for Maxwell’s equations. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford, 2003.
Mor66.
C. Morrey. Multiple Integrals in the Calculus of Variations. Springer-Verlag, New York, 1966.
MPI09.
MS07a.
J. Mandel and B. Sousedík. Adaptive selection of face coarse degrees of freedom in the BDDC and FETI-DP iterative substructuring methods. Comput. Methods Appl. Mech. Engrg., 196(8):1389–1399, 2007.
MS07b.
J. Mandel and B. Sousedík. BDDC and FETI-DP under minimalist assumptions. Computing, 81(4):269–280, 2007.
MSD08.
J. Mandel, B. Sousedík, and C. R. Dohrmann. Multispace and multilevel BDDC. Computing, 83(2-3):55–85, 2008.
MSŠ12.
MT96.
J. Mandel and R. Tezaur. Convergence of a substructuring method with Lagrange multipliers. Numer. Math., 73(4):473–487, 1996.
MT01.
J. Mandel and R. Tezaur. On the convergence of a dual-primal substructuring method. Numer. Math., 88(3):543–558, 2001.
MTF99.
J. Mandel, R. Tezaur, and C. Farhat. A scalable substructuring method by Lagrange multipliers for plate bending problems. SIAM J. Numer. Anal., 36(5):1370–1391, 1999.
Muc72.
B. Muckenhoupt. Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc., 165:207–226, 1972.
Neč67.
J. Nečas. Les méthodes directes en théorie des équations elliptiques. Masson et Cie, Éditeurs, Paris, 1967.
Néd80.
J. C. Nédélec. Mixed finite elements in \({\mathbb{R}}^{3}\). Numer. Math., 35(3):315–341, 1980.
Néd86.
J. C. Nédélec. A new family of mixed finite elements in \({\mathbb{R}}^{3}\). Numer. Math., 50(1):57–81, 1986.
Nep91a.
S. V. Nepomnyaschikh. Mesh theorems on traces, normalizations of function traces and their inversion. Russion Journal of Numerical Analysis and Mathematical Modelling, 6(3):233–242, 1991.
Nep91b.
S. V. Nepomnyaschikh. Method of splitting into subspaces for solving elliptic boundary value problems in complex-form domains. Soviet J. Numer. Anal. Math. Modelling, 6(2):151–168, 1991.
Nep92.
S. V. Nepomnyaschikh. Domain decomposition and ficticious domains methods for elliptic boundary value problems. In Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations, pages 62–72. SIAM, Philadelphia, PA, 1992.
Nep07.
S. Nepomnyaschikh. Domain decomposition methods. In Lectures on advanced computational methods in mechanics, volume 1 of Radon Series in Computational and Applied Mathematics, pages 89–159. Walter de Gruyter, Berlin, 2007.
ODRW89.
J. T. Oden, L. Demkowicz, W. Rachowicz, and T. A. Westermann. Toward a universal h-p adaptive finite element strategy. II. A posteriori error estimation. Comput. Methods Appl. Mech. Engrg., 77(1-2):113–180, 1989.
Of06.
G. Of. BETI-Gebietszerlegungsmethoden mit schnellen Randelementverfahren und Anwendungen. PhD thesis, Universität Stuttgart, Germany, January 2006.
Of08.
G. Of. The all-floating BETI method: Numerical results. In U. Langer, M. Discacciati, D. E. Keyes, O. B. Widlund, and W. Zulehner, editors, Domain Decomposition Methods in Science and Engineering XVII, volume 60 of Lecture Notes in Computational Science and Engineering, pages 295–302. Springer-Verlag, Berlin, 2008.
OS09.
G. Of and O. Steinbach. The all-floating boundary element tearing and interconnecting method. J. Num. Math., 17(4):277–298, 2009.
OSU06.
G. Of, O. Steinbach, and P. Urthaler. Fast evaluation of Newton potentials in the boundary element method. Technical report no. 2008/3, Department for Numerical Mathematics, Graz University of Technology, Austria, 2006.
Osw99.
P. Oswald. On the robustness of the BPX-preconditioner with respect to jumps in the coefficients. Math. Comp., 68(226):633–650, 1999.
PAR05.
Pav97.
L. F. Pavarino. Neumann-Neumann algorithms for spectral elements in three dimensions. RAIRO Modél. Math. Anal. Numér., 31:471–493, 1997.
Pav07.
L. F. Pavarino. BDDC and FETI-DP preconditioners for spectral element discretizations. Comput. Methods Appl. Mech. Engrg., 196(8):1380–1388, 2007.
Pec08a.
C. Pechstein. BETI-DP methods in unbounded domains. In K. Kunisch, G. Of, and O. Steinbach, editors, Numerical Mathematics and Advanced Applications – ENUMATH 2007, pages 381–388. Springer-Verlag, Berlin, 2008.
Pec08b.
Pec09.
C. Pechstein. Boundary element tearing and interconnecting methods in unbounded domains. Appl. Numer. Math., 59(11):2824–2842, 2009.
Pec12.
Pet89.
T. von Petersdorff. Randwertprobleme der Elastizitätstheorie für Polyeder – Singularitäten und Approximation mit Randelementmethoden. PhD thesis, Technische Hochschule Darmstadt, Germany, 1989.
Pie00.
K. H. Pierson. A Family of Domain Decomposition Methods for the Massively Parallel Solution of Computational Mechanics Problems. PhD thesis, Aerospace Engineering, University of Colorado at Boulder, Boulder, CO, 2000.
PJF97.
K. C. Park, M. R. Justino, Jr., and C. A. Felippa. An algebraically partitioned FETI method for parallel structural analysis: algorithm description. Int. J. Numer. Meth. Engng., 40(15):2717–2737, 1997.
PS72.
C. C. Paige and M. A. Saunders. Computational variants of the Lanczos method for eigenproblems. J. Numer. Anal., 15:801–812, 1972.
PS08.
C. Pechstein and R. Scheichl. Analysis of FETI methods for multiscale PDEs. Numer. Math., 111(2):293–333, 2008.
PS09.
C. Pechstein and R. Scheichl. Scaling up through domain decomposition. Appl. Anal., 88(10–11):1589–1608, 2009.
PS10.
C. Pechstein and R. Scheichl. Robust FETI solvers for multiscale elliptic PDEs. In J. Roos and L. R. J. Costa, editors, Scientific Computing in Electrical Engineering SCEE 2008, volume 14 of Mathematics in Industry. Springer-Verlag, Berlin, 2010.
PS11a.
A. Pechstein and J. Schöberl. Tangential-displacement and normal-normal-stress continuous mixed finite elements for elasticity. Math. Models Methods Appl. Sci., 21(8):1761–1782, 2011.
PS11b.
C. Pechstein and R. Scheichl. Analysis of FETI methods for multiscale PDEs. Part II: interface variation. Numer. Math., 118(3):485–529, 2011.
PS11c.
C. Pechstein and R. Scheichl. Weighted Poincaré inequalities and applications in domain decomposition. In Y. Huang, R. Kornhuber, O. Widlund, and J. Xu, editors, Decomposition Methods in Science and Engineering XIX, volume 78 of Lecture Notes in Computational Science and Engineering, pages 197–204. Springer-Verlag, Berlin, 2011.
PS12a.
A. Pechstein and J. Schöberl. Anisotropic mixed finite elements for elasticity. Int. J. Numer. Meth. Engng., 90(2):196–217, 2012.
PS12b.
C. Pechstein and R. Scheichl. Weighted Poincaré inequalities. IMA J. Numer. Anal., 2012. to appear.
PSS12.
PW60.
L. E. Payne and H. F. Weinberger. An optimal Poincaré inequality for convex domains. Arch. Rat. Mech. Anal., 5:286–292, 1960.
PW02.
L. F. Pavarino and O. B. Widlund. Balancing Neumann-Neumann methods for incompressible Stokes equations. Comm. Pure Appl. Math., 55(3):302–335, 2002.
PWZ10.
L. F. Pavarino, O. B. Widlund, and S. Zampini. BDDC preconditioners for spectral element discretizations of almost incompressible elasticity in three dimensions. SIAM J. Sci. Comput., 32(6):3604–3626, 2010.
QV99.
A. Quarteroni and A. Valli. Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications, Oxford, 1999.
RF98a.
D. Rixen and C. Farhat. Preconditioning the FETI method for problems with intra- and inter-subdomain coefficient jumps. In P. E. Bjørstad, M. Espedal, and D. E. Keyes, editors, Proceedings of 9th International Conference on Domain Decomposition, pages 472–479, 1998. http://​www.​ddm.​org/​DD9/​Rixen.​pdf.
RF98b.
F.-X. Roux and C. Farhat. Parallel implementation of direct solution strategies for the coarse grid solvers in 2-level FETI method. In J. Mandel, C. Farhat, and X. Cai, editors, Domain Decomposition Methods 10. Proceedings of the 10th International Conference held in Boulder, CO, August 10–14, 1997, volume 218 of Contemporary Mathematics, pages 158–173. AMS, Providence, RI, 1998. http://​www.​ddm.​org/​DD10/​.
RF99.
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(4):489–516, 1999.
RFTM99.
D. Rixen, C. Farhat, R. Tezaur, and J. Mandel. Theoretical comparison of the FETI and algebraically partitioned FETI methods, and performance comparisons with a direct sparse solver. Internat. J. Numer. Methods Engrg., 46(4):501–533, 1999.
Rhe02.
Oliver Rheinbach. FETI – A dual iterative substructuring method for elliptic partial differential equations. Master’s thesis, Mathematisches Institut, Universität zu Köln, Germany, 2002.
Rhe06.
O. Rheinbach. Parallel Scalable Iterative Substructuring: Robust Exact and Inexact FETI-DP Methods with Applications to Elasticity. PhD thesis, Universität Essen-Duisburg, 2006.
Riv08.
B. Rivière. Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, volume 35 of Frontiers in Applied Mathematics. SIAM, Philadelphia, PA, 2008.
Rix97.
D. Rixen. Substructuring and dual methods in structural analysis. PhD thesis, Collection des Publications de la Faculté des Sciences Appliquées no. 175, Université de Liège, Belgium, 1997.
Rja90.
S. Rjasanow. Vorkonditionierte iterative Auflösung von Randelementgleichungen für die Dirichlet-Aufgabe. Wissenschaftliche Schriftenreihe der Technischen Universität Chemnitz, Chemnitz, 1990.
ROD89.
W. Rachowicz, J. T. Oden, and L. Demkowicz. Toward a universal h-p adaptive finite element strategy. III. Design of h-p meshes. Comput. Methods Appl. Mech. Engrg., 77(1-2):181–212, 1989.
Rog06.
L. G. Rogers. Degree-independent Sobolev extension on locally uniform domains. J. Funct. Anal., 235(2):619–665, 2006.
Rok85.
V. Rokhlin. Rapid solution of integral equations of classical potential theory. J. Comput. Phys., 60(2):187–207, 1985.
Rou09.
F.-X. Roux. A FETI-2LM method for non-matching grids. In Domain Decomposition Methods in Science and Engineering XVIII, volume 70 of Lecture Notes in Computational Science and Engineering, pages 121–128. Springer-Verlag, Heidelberg, 2009.
RS85.
J. Ruge and K. Stüben. Efficient solution of finite difference and finite element equations by algebraic multigrid (AMG). In D. J. Paddon and H. Holstein, editors, Multigrid Methods for Integral and Differential Equations, IMA Conference Series, pages 169–212. Clarendon Press, Oxford, 1985.
RS07.
S. Rjasanow and O. Steinbach. The Fast Solution of Boundary Integral Equations. Mathematical and Analytical Techniques with Applications to Engineering. Springer, New York, 2007.
RT77.
P. A. Raviart and J. M. Thomas. Primal hybrid finite element methods for 2nd order elliptic equations. Math. Comp., 31(138):391–413, 1977.
RT01.
F. Rapetti and A. Toselli. A FETI preconditioner for two-dimensional edge element approximations of Maxwell’s equations on nonmatching grids. SIAM J. Sci. Comput., 23(1):92–108, 2001.
Saa03.
Y. Saad. Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia, 2003.
Sar93.
M. V. Sarkis. Two-level Schwarz methods for nonconforming finite elements and discontinuous coefficients. In N. D. Melson, T. A. Manteuffel, and S. F. McCormick, editors, Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, volume 2, pages 543–566. NASA, Hampton, VA, 1993.
Sar94.
M. V. Sarkis Martins. Schwarz Preconditioners for Elliptic Problems with Discontinuous Coefficients Using Conforming and Non-Conforming Elements. Technical report, Courant Institute of Mathematical Sciences, 1994. TR-671, Department of Computer Science.
Sar97.
M. Sarkis. Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements. Numer. Math., 77(3):383–406, 1997.
SBG96.
B. F. Smith, P. E. Bjørstad, and W. Gropp. Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, 1996.
ŠČBN12.
Sch98a.
R. Schneider. Multiskalen- und Wavelet-Matrixkompression. Advances in Numerical Mathematics. B. G. Teubner, Stuttgart, 1998.
Sch98b.
Ch. Schwab. p- and hp-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. Oxford University Press, New York, 1998.
SDH04.
E. Stein, R. De Borst, and T. J. R. Hughes. Encyclopedia of Computational Mechanics, Volume 2: Solids and Structures. John Wiley & Sons, 2004.
SDH+11.
N. Spillane, V. Dolean, P. Hauret, F. Nataf, C. Pechstein, and R. Scheichl. Abstract robust coarse spaces for systems of PDEs via gereralized eigenproblems in the overlap. NuMa-Report 2011-07, Institute of Computational Mathematics, Johannes Kepler University Linz, 2011.
SG04.
O. Schenk and K. Gärtner. Solving unsymmetric sparse systems of linear equations with PARDISO. Journal of Future Generation Computer Systems, 20(3):475–487, 2004.
SG06.
O. Schenk and K. Gärtner. On fast factorization pivoting methods for sparse symmetric indefinite systems. Electron. Trans. Numer. Anal., 23:158–179, 2006.
Sin08.
SK98.
SK04.
H.-R. Schwarz and N. Köckler. Numerische Mathematik. Teubner, Stuttgart, 2004.
SM08.
B. Sousedík and J. Mandel. On the equivalence of primal and dual substructuring preconditioners. Electron. Trans. Numer. Anal., 31:384–402, 2008.
ŠMSB12.
ŠNM+10.
J. Šístek, J. Novotný, J. Mandel, M. Čertíková, and P. Burda. BDDC by a frontal solver and the stress computation in a hip joint replacement. Math. Comp. Simulation, 80(6):1310–1323, 2010.
Sob91.
S. L. Sobolev. Some applications of functional analysis in mathematical physics, volume 91 of Translations of Mathematical Monographs. AMS, Providence, RI, 1991. Original Russion version published by Leningrad State University, 1950.
Sou10.
Sou12.
SS11.
S. Sauter and C. Schwab. Boundary Element Methods, volume 39 of Springer series in Compuational Mathematics. Springer-Verlag, Berlin, 2011.
ŠSB+11.
J. Šístek, B. Sousedík, P. Burda, J. Mandel, and J. Novotný. Application of the parallel BDDC preconditioner to the Stokes flow. Comput. & Fluids, 46(1):429–435, 2011.
ST98.
E. P. Stephan and T. Tran. Domain decomposition algorithms for indefinite hypersingular integral equations: the h and p versions. SIAM J. Sci. Comput., 19(4):1139–1153 (electronic), 1998.
Ste70.
E. M. Stein. Singular Integrals and Differentiability Properties of Functions, volume 30 of Princeton Math Series. Princeton University Press, Princeton, New Jersey, 1970.
Ste00.
O. Steinbach. OSTBEM – A boundary element software package. Technical report, University of Stuttgart, 2000.
Ste01.
D. Stefanica. A numerical study of FETI algorithms for mortar finite element methods. SIAM J. Sci. Comput., 23(4):1135–1160, 2001.
Ste03a.
O. Steinbach. Artificial multilevel boundary element preconditioners. Proc. Appl. Math. Mech., 3:539–542, 2003.
Ste03b.
O. Steinbach. Stability estimates for hybrid coupled domain decomposition methods, volume 1809 of Lecture Notes in Mathematics. Springer-Verlag, Heidelberg, 2003.
Ste08.
O. Steinbach. Numerical Approximation Methods for Elliptic Boundary Value Problems. Finite and Boundary Elements. Springer-Verlag, New York, 2008.
Ste11.
O. Steinbach. A note on the stable one-equation coupling of finite and boundary elements. SIAM J. Numer. Anal., 49(4):1521–1531, 2011.
SV07.
R. Scheichl and E. Vainikko. Additive Schwarz with aggregation-based coarsening for elliptic problems with highly variable coefficients. Computing, 80(4):319–343, 2007.
SVZ11.
R. Scheichl, P. S. Vassilevski, and L. T. Zikatanov. Weak approximation properties of elliptic projections with functional constraints. Multiscale Modeling and Simulation and Analysis, 9(4):1677–1699, 2011.
SVZ12.
R. Scheichl, P. S. Vassilevski, and L. T. Zikatanov. Mutilevel methods for elliptic problems with highly varying coefficients on non-aligned coarse grids. SIAM J. Numer. Anal., 2012. to appear, Preprint LLNL-JRNL-404462, Lawrence Livermore National Laboratory, CA.
SW01.
O. Steinbach and W. L. Wendland. On C. Neumann’s method for second-order elliptic systems in domains with non-smooth boundaries. J. Math. Anal. Appl., 262(2):733–748, 2001.
SW11a.
O. Steinbach and M. Windisch. Robust boundary element domain decomposition solvers in acoustics. In Y. Huang, R. Kornhuber, O. Widlund, and J. Xu, editors, Domain Decomposition Methods in Science and Engineering XIX, volume 78 of Lecture Notes in Computational Science and Engineering, pages 277–284. Springer-Verlag, Heidelberg, 2011.
SW11b.
O. Steinbach and M. Windisch. Stable boundary element domain decomposition methods for the Helmholtz equation. Numer. Math., 118(1):171–195, 2011.
SW12.
O. Steinbach and M. Windisch. Stable BETI methods in electromagnetics. In R. Bank, R. Kornhuber, and O. Widlund, editors, Domain Decomposition Methods in Science and Engineering XX, Lecture Notes in Computational Science and Engineering. Springer-Verlag, 2012. to appear.
SZ90.
L. R. Scott and S. Zhang. Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comp., 54:483–493, 1990.
Tez98.
TK01.
A. Toselli and A. Klawonn. A FETI domain decomposition method for edge element approximations in two dimensions with discontinous coefficients. SIAM J. Numer. Anal., 39(3):932–956, 2001.
TL08.
X. Tu and J. Li. A balancing domain decomposition method by constraints for advection-diffusion problems. Comm. App. Math. and Comp. Sci., 3(1), 2008.
Tos99.
A. Toselli. Domain Decomposition for Vector Field Problems. PhD thesis, Courant Institute of Mathematical Sciences, New York University, New York, USA, May 1999.
Tos00.
A. Toselli. Neumann-Neumann methods for vector field problems. Electron. Trans. Numer. Anal., 11:1–24, 2000.
Tos01.
A. Toselli. FETI domain decomposition methods for scalar advection-diffusion problems. Comput. Methods Appl. Mech. Engrg., 190(43–44):5759–5776, 2001.
Tos06.
A. Toselli. Dual-primal FETI algorithms for egde finite-element approximations in 3D. IMA J. Numer. Anal., 26:96–130, 2006.
Tu05.
X. Tu. A BDDC algorithm for a mixed formulation of flow in porous media. Electron. Trans. Numer. Anal., 20:164–179, 2005.
Tu07a.
X. Tu. A BDDC algorithm for flow in porous media with a hybrid finite element discretization. Electron. Trans. Numer. Anal., 26:146–160, 2007.
Tu07b.
X. Tu. Three-level BDDC in three dimensions. SIAM J. Sci. Comp., 29(4):1759–1780, 2007.
Tu07c.
X. Tu. Three-level BDDC in two dimensions. Int. J. Numer. Meth. Engng., 69(1):33–59, 2007.
Tu11.
X. Tu. A three-level BDDC algorithm for saddle point problems. Numer. Math., 119:189–217, 2011.
TV03.
A. Toselli and X. Vasseur. A numerical study on Neumann-Neumann and FETI methods for hp-approximations on geometrically refined boundary layer meshes in two dimensions. Comput. Methods Appl. Mech. Engrg., 192:4551–4579, 2003.
TV04.
A. Toselli and X. Vasseur. A numerical study on Neumann-Neumann and FETI methods for hp-approximations boundary layer meshes in three dimensions. IMA J. Numer. Anal., 24(1):123–156, 2004.
TV06.
A. Toselli and X. Vasseur. A numerical study on Neumann-Neumann methods for hp approximations on geometrically refined boundary layer meshes ii. Three-dimensional problems. ESAIM Math. Model. Numer. Anal., 40(1):99–122, 2006.
TW05.
A. Toselli and O. B. Widlund. Domain Decomposition Methods – Algorithms and Theory, volume 34 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2005.
Urt08.
Van82.
H. Van der Vorst. A generalized Lanczos scheme. Math. Comp., 39:559–561, 1982.
Vas08.
P. S. Vassilevski. Multilevel block factorization preconditioners. Springer-Verlag, New York, 2008.
VMB96.
P. Vaněk, J. Mandel, and M. Brezina. Algebraic multigrid by smoothed aggregation for 2nd and 4th order elliptic problems. Computing, 56(3):179–196, 1996.
VSM99.
C. Vuik, A. Segal, and J. A. Meijerink. An efficient preconditioned CG method for the solution of a class of layered problems with extreme contrasts in the coefficients. J. Comput. Phys., 152(1):385–403, 1999.
VSMW01.
C. Vuik., A. Segal., J. A. Meijerink, and G. T. Wijma. The construction of projection vectors for a deflated ICCG method applied to problems with extreme contrasts in the coefficients. J. Comput. Phys., 172(2):426–450, 2001.
VV09.
A. Veeser and R. Verfürth. Explicit upper bounds for dual norms of residuals. SIAM J. Numer. Anal., 47(3):2387–2405, 2009.
VV11.
A. Veeser and R. Verfürth. Poincaré constants of finite element stars. IMA J. Numer. Anal., 32(1):40–47, 2011.
Wei98.
Win10.
WX94.
J. Wang and R. Xie. Domain decomposition for elliptic problems with large jumps in coefficients. In Proceedings of Conference on Scientific and Engineering Computing, pages 74–86. National Defense Industry Press, Beijing, China, 1994.
XZ08.
J. Xu and Y. Zhu. Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients. Math. Mod. Meth. Appl. Sci., 18(1):77–105, 2008.
Zag06.
Zei90.
E. Zeidler. Nonlinear Functional Analysis and Its Applications – Part 2B: Nonlinear Monotone Operators. Springer-Verlag, New York, 1990.
ZKB77.
O. C. Zienkiewicz, D. M. Kelly, and P. Bettess. The coupling of the finite element method and boundary solution procedures. Int. J. Numer. Meth. Eng., 11(2):355–375, 1977.
ZKB79.
O. C. Zienkiewicz, D. M. Kelly, and P. Bettess. Marriage a la mode – the best of both worlds (Finite elements and boundary integrals). In R. Glowinski, E. Y. Rodin, and O. C. Zienkiewicz, editors, Energy Methods in Finite Element Analysis, chapter 5, pages 81–106. Wiley and Son, London, 1979.
ZP08.
V. V. Zhikov and S. E. Pastukhova. Homogenization of degenerate elliptic equations. Sib. Math. J., 49(1):80–101, 2008.
Zul02.
W. Zulehner. Analysis of iterative methods for saddle point problems: A unified approach. Mathematics of Computation, 71:479–505, 2002.
Zul11.
W. Zulehner. Nonstandard norms and robust estimates for saddle point problems. SIAM J. Matrix Anal. Appl., 32(2):536–560, 2011.
Zum03.
G. Zumbusch. Parallel Multilevel Methods – Adaptive Mesh Refinement and Loadbalancing. B. G. Teubner, Stuttgart, Leipzig, Wiesbaden, 2003.
Metadaten
Titel
Preliminaries
verfasst von
Clemens Pechstein
Copyright-Jahr
2013
Verlag
Springer Berlin Heidelberg
Buch
Finite and Boundary Element Tearing and Interconnecting Solvers for Multiscale Problems

Print ISBN: 978-3-642-23587-0

Electronic ISBN: 978-3-642-23588-7

Copyright-Jahr: 2013

DOI
https://doi.org/10.1007/978-3-642-23588-7_1