Abstract
Different parallel two-level overlapping Schwarz preconditioners with Generalized Dryja–Smith–Widlund (GDSW) and Reduced dimension GDSW (RGDSW) coarse spaces for elasticity problems are considered. GDSW type coarse spaces can be constructed from the fully assembled system matrix, but they additionally need the index set of the interface of the corresponding nonoverlapping domain decomposition and the null space of the elasticity operator, i.e., the rigid body motions. In this paper, fully algebraic variants, which are constructed solely from the uniquely distributed system matrix, are compared to the classical variants which make use of this additional information; the fully algebraic variants use an approximation of the interface and an incomplete algebraic null space. Nevertheless, the parallel performance of the fully algebraic variants is competitive compared to the classical variants for a stationary homogeneous model problem and a dynamic heterogenous model problem with coefficient jumps in the shear modulus; the largest parallel computations were performed on 4096 MPI (Message Passing Interface) ranks. The parallel implementations are based on the Trilinos package FROSch.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
X.-C. Cai and M. Sarkis. A restricted additive Schwarz preconditioner for general sparse linear systems. SIAM Journal on Scientific Computing, 21:239–247, 1999.
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.
C. R. Dohrmann, A. Klawonn, and O. B. Widlund. A family of energy minimizing coarse spaces for overlapping Schwarz preconditioners. In Domain decomposition methods in science and engineering XVII, volume 60 of Lect. Notes Comput. Sci. Eng., pages 247–254. Springer, Berlin, 2008.
C. R. Dohrmann and O. B. Widlund. Hybrid domain decomposition algorithms for compressible and almost incompressible elasticity. Internat. J. Numer. Meth. Engng, 82(2):157–183, 2010.
C. R. Dohrmann and O. B. Widlund. On the design of small coarse spaces for domain decomposition algorithms. SIAM J. Sci. Comput., 39(4):A1466–A1488, 2017.
S. C. Eisenstat and H. F. Walker. Choosing the forcing terms in an inexact Newton method. volume 17, pages 16–32. 1996. Special issue on iterative methods in numerical linear algebra (Breckenridge, CO, 1994).
A. Heinlein, C. Hochmuth, and A. Klawonn. Reduced dimension GDSW coarse spaces for monolithic Schwarz domain decomposition methods for incompressible fluid flow problems. International Journal for Numerical Methods in Engineering, 121(6):1101–1119, 2020.
A. Heinlein, A. Klawonn, J. Knepper, and O. Rheinbach. Adaptive GDSW coarse spaces for overlapping Schwarz methods in three dimensions. SIAM Journal on Scientific Computing, 41(5):A3045–A3072, 2019.
A. Heinlein, A. Klawonn, S. Rajamanickam, and O. Rheinbach. FROSch – a parallel implementation of the GDSW domain decomposition preconditioner in Trilinos. In preparation.
A. Heinlein, A. Klawonn, S. Rajamanickam, and O. Rheinbach. FROSch: A Fast and Robust Overlapping Schwarz Domain Decomposition Preconditioner Based on Xpetra in Trilinos. Technical report, Universität zu Köln, November 2018.
A. Heinlein, A. Klawonn, and O. Rheinbach. A parallel implementation of a two-level overlapping Schwarz method with energy-minimizing coarse space based on Trilinos. SIAM J. Sci. Comput., 38(6):C713–C747, 2016.
A. Heinlein, A. Klawonn, O. Rheinbach, and O. B. Widlund. Improving the parallel performance of overlapping Schwarz methods by using a smaller energy minimizing coarse space. International Conference on Domain Decomposition Methods, pages 383–392. Springer, 2017.
M. A. Heroux, R. A. Bartlett, V. E. Howle, R. J. Hoekstra, J. J. Hu, T. G. Kolda, R. B. Lehoucq, K. R. Long, R. P. Pawlowski, E. T. Phipps, A. G. Salinger, H. K. Thornquist, R. S. Tuminaro, J. M. Willenbring, A. Williams, and K. S. Stanley. An overview of the Trilinos project. ACM Trans. Math. Softw., 31(3):397–423, 2005.
Acknowledgements
The authors gratefully acknowledge financial support by the German Science Foundation (DFG), project no. 214421492 and the computing time granted by the Center for Computational Sciences and Simulation (CCSS) of the University of Duisburg-Essen and provided on the supercomputer magnitUDE (DFG grants INST 20876/209-1 FUGG, INST 20876/243-1 FUGG) at the Zentrum für Informations- und Mediendienste (ZIM). We further thank Jascha Knepper for providing the foam geometry.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Heinlein, A., Hochmuth, C., Klawonn, A. (2021). Fully Algebraic Two-Level Overlapping Schwarz Preconditioners for Elasticity Problems. In: Vermolen, F.J., Vuik, C. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2019. Lecture Notes in Computational Science and Engineering, vol 139. Springer, Cham. https://doi.org/10.1007/978-3-030-55874-1_52
Download citation
DOI: https://doi.org/10.1007/978-3-030-55874-1_52
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-55873-4
Online ISBN: 978-3-030-55874-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)