View source for Domain decomposition methods for waves

{{metamodelGetDP|ddm_waves}}

== Additional information ==

To run the models, open '''main.pro''' with Gmsh. (This is mainly for demonstration purposes. For actual, parallel computations, you should recompile GetDP and Gmsh with MPI support, and run from the command line: see [[GetDDM]] for detailed installation instructions).

The formulations implement non-overlapping Schwarz domain decomposition methods for the Helmholtz equation and for the time-harmonic Maxwell system. Several families of transmission conditions are implemented: zeroth- and second-order optimized conditions<ref name=Despres1 /><ref name=Despres2 /><ref name=Gander1 /><ref name=Gander2 /><ref name=Boubendir1 /><ref name=RawatL10 />, new Padé-localized square-root conditions<ref name=BoubAntGeuz2012 /><ref name=ElBouAntGeuz2014 /> and PML conditions. Several variants of the recently proposed double-sweep preconditioner<ref name=VionGeuz2014 /> are also implemented.

For more information about these methods as well as the implementation, please refer to the following preprint: [http://www.montefiore.ulg.ac.be/~geuzaine/preprints/getddm_preprint.pdf GetDDM: an Open Framework for Testing Optimized Schwarz Methods for Time-Harmonic Wave Problems] <ref name=GetDDM />.

== References ==

<references>

<ref name=Despres1>B. Després, Méthodes de Décomposition de Domaine pour les Problèmes de Propagation d'Ondes en Régime Harmonique. Le Théorème de Borg pour l'Equation de Hill Vectorielle, PhD Thesis, Paris VI University, France, 1991.</ref>

<ref name=Despres2>B. Després, P. Joly and J. Roberts, A domain decomposition method for the harmonic Maxwell equations, Iterative methods in linear algebra (Brussels, 1991), pp. 475-484, North-Holland, 1992.</ref>

<ref name=Gander1>M. Gander, F. Magoulès and F. Nataf, Optimized Schwarz methods without overlap for the Helmholtz equation},
SIAM Journal on Scientific Computing, 24(1), pp. 38-60, 2002.</ref>

<ref name=Gander2>V. Dolean, M. Gander and L. Gerardo-Giorda, Optimized Schwarz methods for Maxwell's equations,
SIAM Journal on Scientific Computing, 31(3), pp. 2193-2213, 2009.</ref>

<ref name=Boubendir1>A. Bendali and Y. Boubendir, Non-Overlapping Domain Decomposition Method for a Nodal Finite
Element Method, Numerische Mathematik 103(4), pp.515-537,  (2006).</ref>

<ref name=RawatL10>V. Rawat and J.-F. Lee, Nonoverlapping Domain Decomposition with Second Order Transmission Condition for the Time-Harmonic Maxwell's Equations, SIAM Journal on Scientific Computing, 32(6), pp. 3584-3603, 2010.</ref>

<ref name=BoubAntGeuz2012>Y. Boubendir, X. Antoine and C. Geuzaine.  [http://www.montefiore.ulg.ac.be/~geuzaine/preprints/ddm_helmholtz_preprint.pdf A quasi-optimal non-overlapping domain decomposition algorithm for the Helmholtz equation]. Journal of Computational Physics 231 (2), 262-280, 2012.</ref>

<ref name=ElBouAntGeuz2014>M. El Bouajaji, X. Antoine and C. Geuzaine. [http://www.montefiore.ulg.ac.be/~geuzaine/preprints/osrc_maxwell_preprint.pdf Approximate local magnetic-to-electric surface operators for time-harmonic Maxwell’s equations]. Journal of Computational Physics 279  241-260, 2014.</ref>

<ref name=VionGeuz2014>A. Vion and C. Geuzaine.  [http://www.montefiore.ulg.ac.be/~geuzaine/preprints/ddm_double_sweep_preprint.pdf Double sweep preconditioner for optimized Schwarz methods applied to the Helmholtz problem]. Journal of Computational Physics 266, 171-190, 2014.</ref>

<ref name=GetDDM>B. Thierry, A.Vion, S. Tournier, M. El Bouajaji, D. Colignon, N. Marsic, X. Antoine, C. Geuzaine. [http://www.montefiore.ulg.ac.be/~geuzaine/preprints/getddm_preprint.pdf GetDDM: an Open Framework for Testing Optimized Schwarz Methods for Time-Harmonic Wave Problems]. 2015.</ref>

</references>


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