Multilevel Spectral Domain Decomposition
Abstract
Highly heterogeneous, anisotropic coefficients, e.g., in the simulation of carbon-Fiber composite components, can lead to extremely challenging finite element systems. Direct solvers for the resulting large and sparse linear systems suffer from severe memory requirements and limited parallel scalability, while iterative solvers in general lack robustness. Two-level spectral domain decomposition methods can provide such robustness for symmetric positive definite linear systems by using coarse spaces based on independent generalized eigenproblems in the subdomains. Rigorous condition number bounds are independent of mesh size, number of subdomains, and coefficient contrast. However, their parallel scalability is still limited by the fact that (in order to guarantee robustness) the coarse problem is solved via a direct method. In this paper, we introduce a multilevel variant in the context of subspace correction methods and provide a general convergence theory for its robust convergence for abstract, elliptic variational problems. Assumptions of the theory are verified for conforming as well as for discontinuous Galerkin methods applied to a scalar diffusion problem. Numerical results illustrate the performance of the method for two- and three-dimensional problems and for various discretization schemes, in the context of scalar diffusion and linear elasticity.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
J. E. Aarnes, Efficient domain decomposition methods for elliptic problems arising from flows in heterogeneous porous media, Comput. Vis. Sci., 8 (2005), pp. 93--106, https://doi.org/10.1007/s00791-005-0155-6.
2.
H. Al Daas and L. Grigori, A class of efficient locally constructed preconditioners based on coarse spaces, SIAM J. Matrix Anal. Appl., 40 (2019), pp. 66--91, https://doi.org/10.1137/18M1194365.
3.
H. Al Daas, L. Grigori, P. Jolivet, and P.-H. Tournier, A multilevel Schwarz preconditioner based on a hierarchy of robust coarse spaces, SIAM J. Sci. Comput., 43 (2021), pp. A1907--A1928, https://doi.org/10.1137/19M1266964.
4.
R. E. Alcouffe, A. Brandt, J. E. Dendy, Jr., and J. W. Painter, The multi-grid method for the diffusion equation with strongly discontinuous coefficients, SIAM J. Sci. Statist. Comput., 2 (1981), pp. 430--454, https://doi.org/10.1137/0902035.
5.
P. Bastian, M. Blatt, A. Dedner, N.-A. Dreier, C. Engwer, R. Fritze, C. Gräser, C. Grüninger, D. Kempf, R. Klöfkorn, M. Ohlberger, and O. Sander, The Dune framework: Basic concepts and recent developments, Comput. Math. Appl., 81 (2021), pp. 75--112, https://doi.org/10.1016/j.camwa.2020.06.007.
6.
P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, R. Kornhuber, M. Ohlberger, and O. Sander, A generic grid interface for parallel and adaptive scientific computing. Part II: Implementation and tests in DUNE, Computing, 82 (2008), pp. 121--138, https://doi.org/10.1007/s00607-008-0004-9.
7.
R. Butler, T. Dodwell, A. Reinarz, A. Sandhu, R. Scheichl, and L. Seelinger, High-performance dune modules for solving large-scale, strongly anisotropic elliptic problems with applications to aerospace composites, Comput. Phys. Commun., 249 (2020), 106997, https://doi.org/10.1016/j.cpc.2019.106997.
8.
Y. Chen, T. A. Davis, W. W. Hager, and S. Rajamanickam, Algorithm $887$: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate, ACM Trans. Math. Software, 35 (2008), 22, https://doi.org/10.1145/1391989.1391995.
9.
M. Christie and M. Blunt, Tenth SPE comparative solution project: A comparison of upscaling techniques, SPE Res. Eval. Engrg., 4 (2001), pp. 308--317, https://doi.org/10.2118/72469-PA.
10.
H. A. Daas, P. Jolivet, and J. Scott, A Robust Algebraic Domain Decomposition Preconditioner for Sparse Normal Equations, preprint, arXiv:2107.09006 [math.NA], 2021, https://arxiv.org/abs/2107.09006.
11.
T. A. Davis, Algorithm 832: UMFPACK v4.3---an unsymmetric-pattern multifrontal method, ACM Trans. Math. Software, 30 (2004), pp. 196--199, https://doi.org/10.1145/992200.992206.
12.
V. Dolean, P. Jolivet, and F. Nataf, An Introduction to Domain Decomposition Methods, SIAM, Philadelphia, 2015, https://doi.org/10.1137/1.9781611974065.
13.
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, Comput. Methods Appl. Math., 12 (2012), pp. 391--414, https://doi.org/10.2478/cmam-2012-0027.
14.
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 (2012), pp. 1175--1199, https://doi.org/10.1051/m2an/2011073.
15.
A. Ern and J. Guermond, Theory and Practice of Finite Element Methods, Springer, Cham, 2004.
16.
A. Ern, A. F. Stephansen, and P. Zunino, A discontinuous Galerkin method with weighted averages for advection--diffusion equations with locally small and anisotropic diffusivity, IMA J. Numer. Anal., 29 (2008), pp. 235--256, https://doi.org/10.1093/imanum/drm050.
17.
J. Galvis and Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high-contrast media, Multiscale Model. Simul., 8 (2010), pp. 1461--1483, https://doi.org/10.1137/090751190.
18.
I. G. Graham, P. O. Lechner, and R. Scheichl, Domain decomposition for multiscale PDEs, Numer. Math., 106 (2007), pp. 589--626, https://doi.org/10.1007/s00211-007-0074-1.
19.
G. Karypis and V. Kumar, Multilevel $k$-way partitioning scheme for irregular graphs, J. Parallel Distrib. Comput., 48 (1998), pp. 96--129, https://doi.org/10.1006/jpdc.1997.1404.
20.
R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users' Guide, SIAM, Philadelphia, 1998, https://doi.org/10.1137/1.9780898719628.
21.
J.-Y. L'Excellent, Multifrontal Methods: Parallelism, Memory Usage and Numerical Aspects, habilitation à diriger des recherches, Ecole Normale Supérieure de Lyon, 2012, https://tel.archives-ouvertes.fr/tel-00737751.
22.
C. Ma, R. Scheichl, and T. Dodwell, Novel Design and Analysis of Generalized FE Methods Based on Locally Optimal Spectral Approximations, preprint, arXiv:2103.09545 [math.NA], 2021, https://arxiv.org/abs/2103.09545.
23.
A. Napov and Y. Notay, An algebraic multigrid method with guaranteed convergence rate, SIAM J. Sci. Comput., 34 (2012), pp. A1079--A1109, https://doi.org/10.1137/100818509.
24.
F. Nataf, H. Xiang, and V. Dolean, A two level domain decomposition preconditioner based on local Dirichlet-to-Neumann maps, C. R. Math. Acad. Sci. Paris, 348 (2010), pp. 1163--1167, https://doi.org/10.1016/j.crma.2010.10.007.
25.
C. Pechstein and R. Scheichl, Weighted Poincaré inequalities, IMA J. Numer. Anal., 33 (2013), pp. 652--686, https://doi.org/10.1093/imanum/drs017.
26.
A. Reinarz, T. Dodwell, T. Fletcher, L. Seelinger, R. Butler, and R. Scheichl, Dune-composites -- a new framework for high-performance finite element modelling of laminates, Composite Struct., 184 (2018), pp. 269--278, https://doi.org/10.1016/j.compstruct.2017.09.104.
27.
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., 50 (2012), pp. 1675--1694, https://doi.org/10.1137/100805248.
28.
B. Smith, P. Bjørstad, and W. Gropp, Domain Decomposition -- Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, Cambridge, UK, 1996.
29.
N. Spillane, V. Dolean, P. Hauret, F. Nataf, C. Pechstein, and R. Scheichl, Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps, Numer. Math., 126 (2014), pp. 741--770, https://doi.org/10.1007/s00211-013-0576-y.
30.
N. Spillane, V. Dolean, P. Hauret, F. Nataf, and D. J. Rixen, Solving generalized eigenvalue problems on the interfaces to build a robust two-level FETI method, C. R. Math. Acad. Sci. Paris, 351 (2013), pp. 197--201, https://doi.org/https://doi.org/10.1016/j.crma.2013.03.010.
31.
A. Toselli and O. Widlund, Domain Decomposition Methods -- Algorithms and Theory, Springer, Berlin, 2005.
32.
J. Willems, Robust multilevel methods for general symmetric positive definite operators, SIAM J. Numer. Anal., 52 (2014), pp. 103--124, https://doi.org/10.1137/120865872.
33.
J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev., 34 (1992), pp. 581--613, https://doi.org/10.1137/1034116.
34.
J. Xu and L. Zikatanov, Algebraic multigrid methods, Acta Numer., 26 (2017), p. 591--721, https://doi.org/10.1017/S0962492917000083.
Information & Authors
Information
Published In
Copyright
© 2022, Society for Industrial and Applied Mathematics.
History
Submitted: 15 June 2021
Accepted: 4 October 2021
Published online: 31 January 2022
Keywords
MSC codes
Authors
Funding Information
Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : EXC 2181/1-390900948
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- Algebraic Construction of Adaptive Coarse Spaces for Two-Level Schwarz PreconditionersSIAM Journal on Scientific Computing, Vol. 47, No. 2 | 10 April 2025
- A highly parallelized multiscale preconditioner for Darcy flow in high-contrast mediaJournal of Computational Physics, Vol. 522 | 1 Feb 2025
- Spectral Coarse Spaces for the Substructured Parallel Schwarz MethodJournal of Scientific Computing, Vol. 91, No. 3 | 16 April 2022
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].