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Robust and scalable domain decomposition solvers for unfitted finite element methods. (English) Zbl 1462.65216
Summary: Unfitted finite element methods, e.g., extended finite element techniques or the so-called finite cell method, have a great potential for large scale simulations, since they avoid the generation of body-fitted meshes and the use of graph partitioning techniques, two main bottlenecks for problems with non-trivial geometries. However, the linear systems that arise from these discretizations can be much more ill-conditioned, due to the so-called small cut cell problem. The state-of-the-art approach is to rely on sparse direct methods, which have quadratic complexity and are thus not well suited for large scale simulations. In order to solve this situation, in this work we investigate the use of domain decomposition preconditioners (balancing domain decomposition by constraints) for unfitted methods. We observe that a straightforward application of these preconditioners to the unfitted case has a very poor behavior. As a result, we propose a customization of the classical BDDC methods based on the stiffness weighting operator and an improved definition of the coarse degrees of freedom in the definition of the preconditioner. These changes lead to a robust and algorithmically scalable solver able to deal with unfitted grids. A complete set of complex 3D numerical experiments shows the good performance of the proposed preconditioners.

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F08 Preconditioners for iterative methods
65N20 Numerical methods for ill-posed problems for boundary value problems involving PDEs
65Y05 Parallel numerical computation
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[1] Badia, S.; Nobile, F.; Vergara, C., Fluid-structure partitioned procedures based on Robin transmission conditions, J. Comput. Phys., 227, 7027-7051, (2008) · Zbl 1140.74010
[2] Belytschko, T.; Moës, N.; Usui, S.; Parimi, C., Arbitrary discontinuities in finite elements, Internat. J. Numer. Methods Engrg., 50, 4, 993-1013, (2001) · Zbl 0981.74062
[3] Parvizian, J.; Düster, A.; Rank, E., Finite cell method: h- and p-extension for embedded domain problems in solid mechanics, Comput. Mech., 41, 1, 121-133, (2007) · Zbl 1162.74506
[4] Burman, E.; Claus, S.; Hansbo, P.; Larson, M. G.; Massing, A., Cutfem: discretizing geometry and partial differential equations, Internat. J. Numer. Methods Engrg., 104, 7, 472-501, (2015) · Zbl 1352.65604
[5] Kamensky, D.; Hsu, M.-C.; Schillinger, D.; Evans, J. A.; Aggarwal, A.; Bazilevs, Y.; Sacks, M. S.; Hughes, T. J.R., An immersogeometric variational framework for fluidstructure interaction: application to bioprosthetic heart valves, Comput. Methods Appl. Mech. Engrg., 284, 1005-1053, (2015)
[6] Rudi, J.; Malossi, A. C.I.; Isaac, T.; Stadler, G.; Gurnis, M.; Staar, P. W.J.; Ineichen, Y.; Bekas, C.; Curioni, A.; Ghattas, O., An extreme-scale implicit solver for complex PDEs: highly heterogeneous flow in earth’s mantle, (Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, SC’15, (2015), ACM New York, NY, USA), 5:1-5:12
[7] Gmeiner, B.; Huber, M.; John, L.; Rüde, U.; Wohlmuth, B., A quantitative performance study for Stokes solvers at the extreme scale, J. Comput. Sci., 17, Part 3, 509-521, (2016)
[8] Badia, S.; Martín, A. F.; Principe, J., Multilevel balancing domain decomposition at extreme scales, SIAM J. Sci. Comput., C22-C52, (2016) · Zbl 1334.65217
[9] Klawonn, A.; Lanser, M.; Rheinbach, O., Toward extremely scalable nonlinear domain decomposition methods for elliptic partial differential equations, SIAM J. Sci. Comput., 37, 6, C667-C696, (2015) · Zbl 1329.65294
[10] Menk, A.; Bordas, S. P.A., A robust preconditioning technique for the extended finite element method, Internat. J. Numer. Methods Engrg., 85, 13, 1609-1632, (2011) · Zbl 1217.74128
[11] Berger-Vergiat, L.; Waisman, H.; Hiriyur, B.; Tuminaro, R.; Keyes, D., Inexact Schwarz-algebraic multigrid preconditioners for crack problems modeled by extended finite element methods, Internat. J. Numer. Methods Engrg., 90, 3, 311-328, (2012) · Zbl 1242.74094
[12] Hiriyur, B.; Tuminaro, R.; Waisman, H.; Boman, E.; Keyes, D., A quasi-algebraic multigrid approach to fracture problems based on extended finite elements, SIAM J. Sci. Comput., 34, 2, A603-A626, (2012) · Zbl 1390.74181
[13] de Prenter, F.; Verhoosel, C. V.; van Zwieten, G. J.; van Brummelen, E. H., Condition number analysis and preconditioning of the finite cell method, Comput. Methods Appl. Mech. Engrg., 316, 297-327, (2017), Special Issue on Isogeometric Analysis: Progress and Challenges
[14] Dohrmann, C. R., A preconditioner for substructuring based on constrained energy minimization, SIAM J. Sci. Comput., 25, 1, 246-258, (2003) · Zbl 1038.65039
[15] Badia, S.; Martín, A. F.; Principe, J., FEMPAR: an object-oriented parallel finite element framework, Arch. Comput. Methods Engng., (2017), (in press). arxiv:1708.01773
[16] FEMPAR: Finite Element Multiphysics PARallel solvers https://gitlab.com/fempar/fempar.
[17] Zampini, S., PCBDDC: A class of robust dual-primal methods in petsc, SIAM J. Sci. Comput., 38, 5, S282-S306, (2016) · Zbl 1352.65632
[18] Marco, O.; Sevilla, R.; Zhang, Y.; Ródenas, J. J.; Tur, M., Exact 3d boundary representation in finite element analysis based on Cartesian grids independent of the geometry, Internat. J. Numer. Methods Engrg., 103, 6, 445-468, (2015) · Zbl 1352.65592
[19] Bader, M., Space-Filling Curves: An Introduction With Applications in Scientific Computing, (2012), Springer Science & Business Media, Google-Books-ID: eIe_OdFP0WkC
[20] Karypis, G., METIS and parmetis, (Padua, D., Encyclopedia of Parallel Computing, (2011), Springer US), 1117-1124
[21] Becker, R., Mesh adaptation for Dirichlet flow control via nitsche’s method, Commun. Numer. Methods. Eng., 18, 9, 669-680, (2002) · Zbl 1073.76582
[22] Nitsche, J., Über ein variationsprinzip zur lösung von Dirichlet-problemen bei verwendung von teilräumen, die keinen randbedingungen unterworfen sind, Abh. Math. Semin. Univ. Hambg., 36, 1, 9-15, (1971) · Zbl 0229.65079
[23] Schillinger, D.; Ruess, M., The finite cell method: a review in the context of higher-order structural analysis of cad and image-based geometric models, Arch. Comput. Methods Eng., 22, 3, 391-455, (2014) · Zbl 1348.65056
[24] Saad, Y., (Iterative Methods for Sparse Linear Systems, Other Titles in Applied Mathematics, (2003), Society for Industrial and Applied Mathematics)
[25] Mandel, J.; Dohrmann, C. R., Convergence of a balancing domain decomposition by constraints and energy minimization, Numer. Linear Algebra Appl., 10, 7, 639-659, (2003) · Zbl 1071.65558
[26] Brenner, S. C.; Scott, R., The Mathematical Theory of Finite Element Methods, (2010), Springer, Softcover reprint of hardcover third ed
[27] Badia, S.; Martín, A. F.; Principe, J., Implementation and scalability analysis of balancing domain decomposition methods, Arch. Comput. Methods Eng., 20, 3, 239-262, (2013) · Zbl 1354.65261
[28] Toselli, A.; Widlund, O., Domain Decomposition Methods, (2004), Springer
[29] Klawonn, A.; Widlund, O. B.; Dryja, M., Dual-primal FETI methods for three-dimensional elliptic problems with heterogeneous coefficients, SIAM J. Numer. Anal., 40, 1, 159-179, (2002) · Zbl 1032.65031
[30] C. Pechstein, C.R. Dohrmann, A Unified Framework for Adaptive BDDC Tech. rep., RICAM-Report 2016-20, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenberger Str. 69, 4040, Linz, Austria, 2016.
[31] B. Sousedík, J. Šístek, J. Mandel, Adaptive- Multilevel BDDC and Its Parallel Implementation, 2013. arxiv:1301.0191.
[32] Klawonn, A.; Radtke, P.; Rheinbach, O., FETI-dp methods with an adaptive coarse space, SIAM J. Numer. Anal., 53, 1, 297-320, (2015) · Zbl 1327.65063
[33] Zampini, S.; Keyes, D. E., On the robustness and prospects of adaptive bddc methods for finite element discretizations of elliptic pdes with high-contrast coefficients, (Proceedings of the Platform for Advanced Scientific Computing Conference, PASC’16, (2016), ACM New York, NY, USA), 6:1-6:13
[34] H.H. Kim, E. Chung, J. Wang, BDDC and FETI-DP algorithms with adaptive coarse spaces for three-dimensional elliptic problems with oscillatory and high contrast coefficients, 2016. ArXiv:1606.07560 [Math]. · Zbl 1380.65374
[35] Calvo, J. G.; Widlund, O. B., An adaptive choice of primal constraints for BDDC domain decomposition algorithms, Electron. Trans. Numer. Anal., 45, 524-544, (2016) · Zbl 1357.65295
[36] Beirão Da Veiga, L.; Cho, D.; Pavarino, L. F.; Scacchi, S., Bddc preconditioners for isogeometric analysis, Math. Models Methods Appl. Sci., 23, 06, 1099-1142, (2012) · Zbl 1280.65138
[37] Badia, S.; Martín, A. F.; Principe, J., A highly scalable parallel implementation of balancing domain decomposition by constraints, SIAM J. Sci. Comput., 36, 2, C190-C218, (2014)
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