zbMATH — the first resource for mathematics

Coarse spaces for FETI-DP and BDDC methods for heterogeneous problems: connections of deflation and a generalized transformation-of-basis approach. (English) Zbl 07176709
In the paper, the authors developed a theory for generalized transformation of basis method for the FETI-DP and BDDC methods. The use of such idea enhances the robustnesss. The result of the paper shows that the developed idea is comparable to the traditional method of deflation in terms of the conditional number of the preconditioned matrix. The proposed method can be used for heterogeneous problems and for the use of adaptive coarse spaces. Some numerical examples are used to demonstrate the theoretical results.

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
74E05 Inhomogeneity in solid mechanics
Full Text: DOI Link
[1] L. BEIRÃO DAVEIGA, L. F. PAVARINO, S. SCACCHI, O. B. WIDLUND,ANDS. ZAMPINI,Adaptive selection of primal constraints for isogeometric BDDC deluxe preconditioners, SIAM J. Sci. Comput., 39 (2017), pp. A281-A302. · Zbl 1360.65090
[2] ,Isogeometric BDDC preconditioners with deluxe scaling, SIAM J. Sci. Comput., 36 (2014), pp. A1118- A1139. · Zbl 1320.65047
[3] P. BJØRSTAD, J. KOSTER,ANDP. KRZYZANOWSKI,Domain decomposition solvers for large scale industrial finite element problems, in Applied Parallel Computing. New Paradigms for HPC in Industry and Academia, T. Sørevik, F. Manne, A. H. Gebremedhin, and R. Moe, eds., vol. 1947 of Lect. Notes Comput. Sci., Springer, Berlin, 2001, pp. 373-383.
[4] P. BJØRSTAD ANDP. KRZYZANOWSKI,A flexible 2-level Neumann-Neumann method for structural analysis problems, in Parallel Processing and Applied Mathematics, R. Wyrzykowski, J. Dongarra, M. Paprzycki, and Jerzy Wa´sniewski, eds., vol. 2328 of Lect. Notes Comput. Sci., Springer, Berlin, 2002, pp. 387-394. · Zbl 1057.65516
[5] C. BOVET, A. PARRET-FRÉAUD, N. SPILLANE,ANDP. GOSSELET,Adaptive multipreconditioned FETI: scalability results and robustness assessment, Computers & Structures, 193 (2017), pp. 1-20.
[6] J. G. CALVO,A BDDC algorithm with deluxe scaling forH(curl)in two dimensions with irregular subdomains, Math. Comp., 85 (2016), pp. 1085-1111. · Zbl 1332.65178
[7] J. G. CALVO ANDO. B. WIDLUND,An adaptive choice of primal constraints for BDDC domain decomposition algorithms, Electron. Trans. Numer. Anal., 45 (2016), pp. 524-544. http://etna.ricam.oeaw.ac.at/vol.45.2016/pp524-544.dir/pp524-544.pdf · Zbl 1357.65295
[8] E. T. CHUNG ANDH. H. KIM,A deluxe FETI-DP algorithm for a hybrid staggered discontinuous Galerkin method for H(curl)-elliptic problems, Internat. J. Numer. Meth. Engng., 98 (2014), pp. 1-23. · Zbl 1352.65483
[9] J. M. CROS,A preconditioner for the Schur complement domain decomposition method, in Domain Decomposition Methods in Science and Engineering, I. Herrera, D. E. Keyes, O. B. Widlund, R. Yates, eds., National Autonomous University of Mexico (UNAM), Mexico City, 2003, pp. 373-380.
[10] C. R. DOHRMANN,A preconditioner for substructuring based on constrained energy minimization, SIAM J. Sci. Comput., 25 (2003), pp. 246-258. · Zbl 1038.65039
[11] ,An approximate BDDC preconditioner, Numer. Linear Algebra Appl., 14 (2007), pp. 149-168. · Zbl 1199.65088
[12] C. DOHRMANN ANDC. PECHSTEIN,Modern domain decomposition solvers - BDDC, deluxe scaling, and an algebraic approach, Talk by C. Pechstein at the University Linz, December 2013. http://people.ricam.oeaw.ac.at/c.pechstein/pechstein-bddc2013.pdf
[13] C. R. DOHRMANN ANDO. B. WIDLUND,Some recent tools and a BDDC algorithm for 3D problems inH(curl), in Domain Decomposition Methods in Science and Engineering XX, R. Bank, M. Holst, O. Widlund, and J. Xu, eds., vol. 91 of Lect. Notes Comput. Sci. Eng., Springer, Heidelberg, 2013, pp. 15-25.
[14] V. DOLEAN, F. NATAF, R. SCHEICHL,ANDN. SPILLANE,Analysis of a two-level Schwarz method with coarse spaces based on local Dirichlet-to-Neumann maps, Comput. Methods Appl. Math., 12 (2012), pp. 391-414. · Zbl 1284.65050
[15] Z. DOSTÁL,Conjugate gradient method with preconditioning by projector, Int. J. Comput. Math., 23 (1988), pp. 315-323. · Zbl 0668.65034
[16] ,Projector preconditioning and domain decomposition methods, Appl. Math. Comput, 37 (1990), pp. 75-81. · Zbl 0701.65078
[17] E. EIKELAND, L. MARCINKOWSKI,ANDT. RAHMAN,Overlapping Schwarz methods with adaptive coarse spaces for multiscale problems in 3D, Preprint on arXiv, 2016. https://arxiv.org/abs/1611.00968 · Zbl 1414.65044
[18] C. FARHAT, M. LESOINNE, P. LETALLEC, K. PIERSON,ANDD. 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 · Zbl 1008.74076
[19] C. FARHAT, M. LESOINNE,ANDK. PIERSON,A scalable dual-primal domain decomposition method, Numer. Linear Algebra Appl., 7 (2000), pp. 687-714. · Zbl 1051.65119
[20] Y. FRAGAKIS ANDM. 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 (2003), pp. 3799-3830. · Zbl 1054.74069
[21] J. GALVIS ANDY. EFENDIEV,Domain decomposition preconditioners for multiscale flows in high-contrast media, Multiscale Model. Simul., 8 (2010), pp. 1461-1483. · Zbl 1206.76042
[22] ,Domain decomposition preconditioners for multiscale flows in high contrast media: reduced dimension coarse spaces, Multiscale Model. Simul., 8 (2010), pp. 1621-1644. · Zbl 1381.65029
[23] M. J. GANDER, A. LONELAND,ANDT. RAHMAN,Analysis of a new harmonically enriched multiscale coarse space for domain decomposition methods, Preprint on arXiv, 2015. https://arxiv.org/abs/1512.05285
[24] S. GIPPERT, A. KLAWONN,ANDO. RHEINBACH,A deflation based coarse space in Dual-Primal FETI methods for almost incompressible elasticity, in Numerical Mathematics and Advanced Applications ENUMATH 2013, A. Abdulle, S. Deparis, D. Kressner, F. Nobile, and M. Picasso, eds., vol. 103 of Lect. Notes Comput. Sci. Eng., Springer, Cham, 2015, pp. 573-581. · Zbl 1321.74070
[25] P. GOLDFELD, L. F. PAVARINO,ANDO. B. WIDLUND,Balancing Neumann-Neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity, Numer. Math., 95 (2003), pp. 283-324. · Zbl 1169.65346
[26] A. HEINLEIN, A. KLAWONN, J. KNEPPER,ANDO. RHEINBACH,Multiscale coarse spaces for overlapping Schwarz methods based on the ACMS space in 2d, Tech. Report 09/2016, Fakultät für Mathematik und Informatik, TU Bergakademie Freiberg, Freiberg, 2016. http://tu-freiberg.de/fakult1/forschung/preprints · Zbl 1448.65263
[27] M. JAROŠOVÁ, A. KLAWONN,ANDO. RHEINBACH,Projector preconditioning and transformation of basis in FETI-DP algorithms for contact problems, Math. Comput. Simulation, 82 (2012), pp. 1894-1907. · Zbl 1254.90147
[28] G. KARYPIS ANDV. KUMAR,Metis, unstructured graph partitioning and sparse matrix ordering system. version 2.0, Tech. Report, University of Minnesota, Department of Computer Science, Minneapolis, August 1995.
[29] ,A fast and high quality multilevel scheme for partitioning irregular graphs, SIAM J. Sci. Comput., 20 (1998), pp. 359-392. · Zbl 0915.68129
[30] H. H. KIM ANDE. T. CHUNG,A BDDC algorithm with enriched coarse spaces for two-dimensional elliptic problems with oscillatory and high contrast coefficients, Multiscale Model. Simul., 13 (2015), pp. 571- 593. · Zbl 1317.65090
[31] H. H. KIM, E. CHUNG,ANDJ. WANG,BDDC and FETI-DP preconditioners with adaptive coarse spaces for three-dimensional elliptic problems with oscillatory and high contrast coefficients, J. Comput. Phys., 349 (2017), pp. 191-214. · Zbl 1380.65374
[32] A. KLAWONN, M. KÜHN,ANDO. RHEINBACH,Adaptive coarse spaces for FETI-DP in three dimensions, SIAM J. Sci. Comput., 38 (2016), pp. A2880-A2911. · Zbl 1346.74168
[33] ,Adaptive FETI-DP and BDDC methods with a generalized transformation of basis for heterogeneous problems, Electron. Trans. Numer. Anal., 49 (2018), pp. 1-27. http://etna.ricam.oeaw.ac.at/vol.49.2018/pp1-27.dir/pp1-27.pdf · Zbl 1448.65236
[34] ,FETI-DP and BDDC methods with a transformation of basis for heterogeneous problems: Connections to deflation, Tech. Report 01/2017, Fakultät für Mathematik und Informatik, TU Bergakademie Freiberg, Freiberg, April 2017. http://tu-freiberg.de/fakult1/forschung/preprints.
[35] ,Parallel adaptive FETI-DP using lightweight asynchronous dynamic load balancing, Internat. J. Numer. Methods Engrg., published online Sept. 12, 2019, doi: 10.1002/nme.6237.
[36] A. KLAWONN, M. LANSER,ANDO. RHEINBACH,Toward extremely scalable nonlinear domain decomposition methods for elliptic partial differential equations, SIAM J. Sci. Comput., 37 (2015), pp. C667-C696. · Zbl 1329.65294
[37] A. KLAWONN, P. RADTKE,ANDO. RHEINBACH,FETI-DP methods with an adaptive coarse space, SIAM J. Numer. Anal., 53 (2015), pp. 297-320. · Zbl 1327.65063
[38] ,A comparison of adaptive coarse spaces for iterative substructuring in two dimensions, Electron. Trans. Numer. Anal., 45 (2016), pp. 75-106. http://etna.ricam.oeaw.ac.at/vol.45.2016/pp75-106.dir/pp75-106.pdf · Zbl 1338.65084
[39] A. KLAWONN ANDO. RHEINBACH,A parallel implementation of dual-primal FETI methods for threedimensional linear elasticity using a transformation of basis, SIAM J. Sci. Comput., 28 (2006), pp. 1886-
[40] ,Robust FETI-DP methods for heterogeneous three dimensional elasticity problems, Comput. Methods Appl. Mech. Engrg., 196 (2007), pp. 1400-1414. · Zbl 1173.74428
[41] ,Deflation, projector preconditioning, and balancing in iterative substructuring methods: connections and new results, SIAM J. Sci. Comput., 34 (2012), pp. A459-A484.
[42] A. KLAWONN ANDO. B. WIDLUND,FETI and Neumann-Neumann iterative substructuring methods: connections and new results, Comm. Pure Appl. Math., 54 (2001), pp. 57-90. · Zbl 1023.65120
[43] ,Dual-primal FETI methods for linear elasticity, Comm. Pure Appl. Math., 59 (2006), pp. 1523-1572. ETNA · Zbl 1110.74053
[44] A. KLAWONN, O. B. WIDLUND,ANDM. DRYJA,Dual-primal FETI methods for three-dimensional elliptic problems with heterogeneous coefficients, SIAM J. Numer. Anal., 40 (2002), pp. 159-179. · Zbl 1032.65031
[45] ,Dual-primal FETI methods with face constraints, in Recent Developments in Domain Decomposition Methods, L. F. Pavarino and A. Toselli, eds., vol. 23 of Lect. Notes Comput. Sci. Eng., Springer, Berlin, 2002, pp. 27-40. · Zbl 1009.65069
[46] M. J. KÜHN,Adaptive FETI-DP and BDDC methods for highly heterogeneous elliptic finite element problems in three dimensions, PhD. Thesis, Mathematisch-Naturwissenschaftlichen Fakultät, Universität zu Köln, Köln, 2018.
[47] J. LI ANDO. B. WIDLUND,FETI-DP, BDDC, and block Cholesky methods, Internat. J. Numer. Meth. Engng., 66 (2006), pp. 250-271.
[48] J. MANDEL, C. R. DOHRMANN,ANDR. TEZAUR,An algebraic theory for primal and dual substructuring methods by constraints, Appl. Numer. Math., 54 (2005), pp. 167-193. · Zbl 1076.65100
[49] J. MANDEL ANDB. SOUSEDÍK,Adaptive selection of face coarse degrees of freedom in the BDDC and the FETI-DP iterative substructuring methods, Comput. Methods Appl. Mech. Engrg., 196 (2007), pp. 1389-1399. · Zbl 1173.74435
[50] J. MANDEL, B. SOUSEDÍK,ANDJ. ŠÍSTEK,Adaptive BDDC in three dimensions, Math. Comput. Simulation, 82 (2012), pp. 1812-1831. · Zbl 1255.65225
[51] R. NABBEN ANDC. VUIK,A comparison of deflation and the balancing preconditioner, SIAM J. Sci. Comput., 27 (2006), pp. 1742-1759. · Zbl 1105.65049
[52] F. NATAF, H. XIANG, V. DOLEAN,ANDN. SPILLANE,A coarse space construction based on local Dirichletto-Neumann maps, SIAM J. Sci. Comput., 33 (2011), pp. 1623-1642. · Zbl 1230.65134
[53] R. A. NICOLAIDES,Deflation of conjugate gradients with applications to boundary value problems, SIAM J. Numer. Anal., 24 (1987), pp. 355-365. · Zbl 0624.65028
[54] D.-S. OH, O. B. WIDLUND, S. ZAMPINI,ANDC. R. DOHRMANN,BDDC Algorithms with deluxe scaling and adaptive selection of primal constraints for Raviart-Thomas vector fields, Math. Comp., 87 (2018), pp. 659-692. · Zbl 1380.65065
[55] C. PECHSTEIN ANDC. R. DOHRMANN,A unified framework for adaptive BDDC, Electron. Trans. Numer. Anal., 46 (2017), pp. 273-336. http://etna.ricam.oeaw.ac.at/vol.46.2017/pp273-336.dir/pp273-336.pdf · Zbl 1368.65043
[56] C. PECHSTEIN ANDR. SCHEICHL,Analysis of FETI methods for multiscale PDEs. Part II: interface variation, Numer. Math., 118 (2011), pp. 485-529. · Zbl 1380.65388
[57] O. RHEINBACH,Parallel scalable iterative substructuring: Robust exact and inexact FETI-DP methods with applications to elasticity, PhD. Thesis, Fachbereich Mathematik, Universität Duisburg-Essen, Duisburg, 2006. · Zbl 1227.65002
[58] ,Parallel iterative substructuring in structural mechanics, Arch. Comput. Methods Eng., 16 (2009), pp. 425-463. · Zbl 1179.74157
[59] M. V. SARKISMARTINS,Schwarz preconditioners for elliptic problems with discontinuous coefficients using conforming and non-conforming elements, PhD. Thesis, Department of Mathematics, New York University, New York, 1994.
[60] N. SPILLANE, V. DOLEAN, P. HAURET, F. NATAF, C. PECHSTEIN,ANDR. SCHEICHL,Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps, Numer. Math., 126 (2014), pp. 741-770. · Zbl 1291.65109
[61] N. SPILLANE ANDD. RIXEN,Automatic spectral coarse spaces for robust FETI and BDD algorithms, Internat. J. Numer. Methods Engng., 95 (2013), pp. 953-990. · Zbl 1352.65553
[62] A. TOSELLI ANDO. B. WIDLUND,Domain Decomposition Methods—Algorithms and Theory, Springer, Berlin, 2005.
[63] S. ZAMPINI,PCBDDC: a class of robust dual-primal methods in PETSc, SIAM J. Sci. Comput., 38 (2016), pp. S282-S306. · Zbl 1352.65632
[64] S.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.