Abstract
Some algorithmic aspects of systems of PDEs based simulations can be better clarified by means of symbolic computation techniques. This is very important since numerical simulations heavily rely on solving systems of PDEs. For the large-scale problems we deal with in today’s standard applications, it is necessary to rely on iterative Krylov methods that are scalable (i.e., weakly dependent on the number of degrees on freedom and number of subdomains) and have limited memory requirements.
This work was supported by the PEPS Maths-ST2I SADDLES.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Y. Achdou, P. Le Tallec, F. Nataf, and M. Vidrascu. A domain decomposition preconditioner for an advection-diffusion problem. Comput. Methods Appl. Mech. Engrg, 184:145–170, 2000.
H. Barucq, J. Diaz, and M. Tlemcani. New absorbing layers conditions for short water waves. Journal of Computational Physics, 229(1):58–72, 2010.
F. Chyzak, A. Quadrat, and D. Robertz. Effective algorithms for parametrizing linear control systems over Ore algebras. Appl. Algebra Engrg. Comm. Comput., 16:319–376, 2005.
F. Chyzak, A. Quadrat, and D. Robertz. OreModules: A symbolic package for the study of multidimensional linear systems. In Applications of Time-Delay Systems, volume 352 of LNCIS, pages 233–264. Springer, 2007.
T. Cluzeau, V. Dolean, F. Nataf, A. Quadrat, Symbolic methods for developing new domain decomposition algorithms, INRIA Tehnical Report RR-7953, 2012. http://hal.inria.fr/hal-00694468
A. D. Cox, J. Little, and D. O’Shea. Using Algebraic Geometry, volume 185 of Graduate Texts in Mathematics. Springer, second edition, 2005.
V. Dolean and F. Nataf. A new domain decomposition method for the compressible Euler equations. M2AN Math. Model. Numer. Anal., 40(4):689–703, 2006.
V. Dolean, F. Nataf, and G. Rapin. Deriving a new domain decomposition method for the Stokes equations using the Smith factorization. Math. Comp., 78(266):789–814, 2009.
Ch. Farhat and F.-X. Roux. A Method of Finite Element Tearing and Interconnecting and its Parallel Solution Algorithm. Internat. J. Numer. Methods Engrg., 32:1205–1227, 1991.
P. Gosselet and C. Rey. Non-overlapping domain decomposition methods in structural mechanics. Arch. Comput. Methods Engrg., 13(4):515–572, 2006.
J. Mandel. Balancing domain decomposition. Comm. on Applied Numerical Methods, 9:233–241, 1992.
F. Nataf. A new approach to perfectly matched layers for the linearized Euler system. J. Comput. Phys., 214(2):757–772, 2006.
L.F. Pavarino and O.B. Widlund. Balancing Neumann-Neumann methods for incompressible Stokes equations. Comm. Pure Appl. Math., 55:302–335, 2002.
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.
J.J. Rotman. An Introduction to Homological Algebra. Springer, second edition, 2009.
P. Le Tallec, J. Mandel, and M. Vidrascu. A Neumann-Neumann Domain Decomposition Algorithm for Solving Plate and Shell Problems. SIAM J. Numer. Anal., 35:836–867, 1998.
J.T. Wloka, B. Rowley, and B. Lawruk. Boundary Value Problems for Elliptic Systems. Cambridge University Press, Cambridge, 1995.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cluzeau, T., Dolean, V., Nataf, F., Quadrat, A. (2013). Symbolic Techniques for Domain Decomposition Methods. In: Bank, R., Holst, M., Widlund, O., Xu, J. (eds) Domain Decomposition Methods in Science and Engineering XX. Lecture Notes in Computational Science and Engineering, vol 91. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35275-1_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-35275-1_3
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-35274-4
Online ISBN: 978-3-642-35275-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)