Abstract
Krylov-type methods are widely used in order to accelerate the convergence of Schwarz-type methods in the linear case. Authors in [2] have shown that they accelerate without overhead cost the convergence speed of Schwarz methods for different types of transmission conditions. In the nonlinear context, the well-known class of Newton-Krylov-Schwarz methods (cf. [5]) for steady-state problems or timedependent problems uses the following strategy: time-dependent problems are discretised uniformly in time first and then one proceeds as for steady-state problems, i.e. the nonlinear problem is solved by a Newton method where the linear system at each iteration is solved by a Krylov-type method preconditioned by an algebraic Schwarz method. The major limitation is that NKS methods do not allow different time discretisations in the subdomains since the problem is discretised in time uniformly up from the beginning.
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
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
Bibliography
D. Bennequin, M. Gander, and L. Halpern. A Homographic Best Approximation Problem with Application to Optimized Schwarz Waveform Relaxation. Math. Comp., 78(265):185–223, 2009.
E. Brakkee and P. Wilders. The Influence of Interface Conditions on Convergence of Krylov-Schwarz Domain Decomposition for the Advection-Diffusion Equation. Journal of Scientific Computing, 12: 11–30, 1997.
S. Brunauer, P. H. Emett, and E. Teller. Adsorption of Gases in Multimolecular Layers. Journal American Chemical Society, 60(2): 309–319, 1938.
F. Caetano, L. Halpern, M. Gander, and J. Szeftel. Schwarz waveform relaxation algorithms for semilinear reaction-diffusion. Networks and heterogeneous media, 5(3):487–505, 2010.
X. C. Cai, W. D. Gropp, D. E. Keyes, and M. D. Tidriri. Parallel implicit methods for aerodynamics. In In Keyes, pages 465–470. American Mathematical Society, 1994.
P. Cresta, O. Allix, C. Rey, and S. Guinard. Nonlinear localization strategies for domain decomposition methods: application to post-buckling analyses. CMAME, 196(8):1436–1446, 2007.
R. Eymard, T. Gallouët, and R. Herbin. Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes sushi: a scheme using stabilization and hybrid interfaces. IMA Journal of Numerical Analysis, 30(4):1009–1043, 2010.
M. J. Gander and L. Halpern. Optimized Schwarz waveform relaxation methods for advection reaction diffusion problems. SIAM J. Numer. Anal., 45(2):666–697, 2007.
F. Haeberlein. Time-Space Domain Decomposition Methods for Reactive Transport Applied to CO 2 Geological Storage. PhD Thesis, University Paris 13, 2011.
F. Haeberlein, A. Michel, and L. Halpern. A test case for multi-species reactive-transport in heterogeneous porous media applied to CO2 geological storate. http://www.ljll.math.upmc.fr/mcparis09/Files/haeberlein_poster.pdf, 2009.
J. Pebrel, C. Rey, and P. Gosselet. A nonlinear dual domain decomposition method: application to structural problems with damage. international journal of multiscale computational engineering, 6(3): 251–262, 2008.
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
Haeberlein, F., Halpern, L., Michel, A. (2013). Newton-Schwarz Optimised Waveform Relaxation Krylov Accelerators for Nonlinear Reactive Transport. 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_45
Download citation
DOI: https://doi.org/10.1007/978-3-642-35275-1_45
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)