[en] In this paper we discuss different transmission operators for the non-overlapping Schwarz method which are suited for solving the time-harmonic Helmholtz equation in cavities (i.e. closed domains which do not feature an outgoing wave condition). Such problems are heavily impacted by back-propagating waves which are often neglected when devising optimized transmission operators for the Schwarz method. This work explores new operators taking into account those back-propagating waves and compares them with well-established operators neglecting these contributions. Notably, this paper focuses on the case of rectangular cavities, as the optimal (non-local) transmission operator can be easily determined. Nonetheless, deviations from this ideal geometry are considered as well. In particular, computations of the acoustic noise in a three-dimensional model of the helium vessel of a beamline cryostat with optimized Schwarz schemes are discussed. Those computations show a reduction of 46% in the iteration count, when comparing an operator optimized for cavities with those optimized for unbounded problems.
Disciplines :
Mathematics
Author, co-author :
Marsic, Nicolas ; Technische Universität Darmstadt, Institute for Accelerator Science and Electromagnetic Fields (TEMF), Darmstadt, Germany
Geuzaine, Christophe ; Université de Liège - ULiège > Département d'électricité, électronique et informatique (Institut Montefiore) > Applied and Computational Electromagnetics (ACE)
De Gersem, Herbert; Technische Universität Darmstadt, Institute for Accelerator Science and Electromagnetic Fields (TEMF), Darmstadt, Germany
Language :
English
Title :
Transmission operators for the non-overlapping Schwarz method for solving Helmholtz problems in rectangular cavities
This research project has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project number 445906998 . The work of Nicolas Marsic is also supported by the Graduate School CE within the Centre for Computational Engineering at the Technische Universität Darmstadt . Computational resources have been provided by the Consortium des Équipements de Calcul Intensif (CÉCI), funded by the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under Grant No. 2.5020.11 and by the Walloon Region . The authors would like to express their gratitude to Mr. Anthony Royer for his help with the GmshFEM and GmshDDM frameworks. In addition, the authors are grateful to Ms. Heike Koch, Mr. Achim Wagner, Mr. Dragos Munteanu, Mr. Christian Schmitt, Dr. Wolfgang F.O. Müller and Dr. David Colignon for the administrative and technical support. Finally, the authors would like to thank the anonymous Reviewers, whose comments improved significantly the quality of this work.
Commentary :
37 pages, 23 figures. Changes with respect to the previous version:
i) one more reference (original GMRES paper) and ii) fixing more typos. This
version is published in Computers & Mathematics with Applications
scite shows how a scientific paper has been cited by providing the context of the citation, a classification describing whether it supports, mentions, or contrasts the cited claim, and a label indicating in which section the citation was made.
Bibliography
Ihlenburg, F., Babuška, I., Finite element solution of the Helmholtz equation with high wave number part I: the h-version of the FEM. Comput. Math. Appl. 30:9 (1995), 9–37, 10.1016/0898-1221(95)00144-N.
Ernst, O.G., Gander, M.J., Why it is difficult to solve Helmholtz problems with classical iterative methods. Graham, I.G., Hou, T.Y., Lakkis, O., Scheichl, R., (eds.) Numerical Analysis of Multiscale Problems Lecture Notes in Computational Science and Engineering, vol. 83, 2012, 325–363, 10.1007/978-3-642-22061-6_10.
Ihlenburg, F., Babuška, I., Finite element solution of the Helmholtz equation with high wave number part II: the h-p version of the FEM. SIAM J. Numer. Anal. 34:1 (1997), 315–358, 10.1137/S0036142994272337.
Moiola, A., Spence, E.A., Is the Helmholtz equation really sign-indefinite?. SIAM Rev. 56:2 (2014), 274–312, 10.1137/120901301.
Diwan, G.C., Moiola, A., Spence, E.A., Can coercive formulations lead to fast and accurate solution of the Helmholtz equation?. J. Comput. Appl. Math. 352 (2019), 110–131, 10.1016/j.cam.2018.11.035.
Yannakakis, M., Computing the minimum fill-in is NP-complete. SIAM J. Algebraic Discrete Methods 2:1 (1981), 77–79, 10.1137/0602010.
Marsic, N., De Gersem, H., Demésy, G., Nicolet, A., Geuzaine, C., Modal analysis of the ultrahigh finesse Haroche QED cavity. New J. Phys., 20(4), 2018, 043058, 10.1088/1367-2630/aab6fd.
Després, B., Décomposition de domaine et problème de Helmholtz. C. R. Acad. Sci. 311 (1990), 313–316 https://gallica.bnf.fr/ark:/12148/bpt6k57815213.
Boubendir, Y., An analysis of the BEM-FEM non-overlapping domain decomposition method for a scattering problem. J. Comput. Appl. Math. 204:2 (2007), 282–291, 10.1016/j.cam.2006.02.044.
Gander, M.J., Magoulès, F., Nataf, F., Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput. 24:1 (2002), 38–60, 10.1137/S1064827501387012.
Boubendir, Y., Antoine, X., Geuzaine, C., A quasi-optimal non-overlapping domain decomposition algorithm for the Helmholtz equation. J. Comput. Phys. 231:2 (2012), 262–280, 10.1016/j.jcp.2011.08.007.
Vion, A., Geuzaine, C., Double sweep preconditioner for optimized Schwarz methods applied to the Helmholtz problem. J. Comput. Phys. 266 (2014), 171–190, 10.1016/j.jcp.2014.02.015.
Gander, M.J., Zhang, H., A class of iterative solvers for the Helmholtz equation: factorizations, sweeping preconditioners, source transfer, single layer potentials, polarized traces, and optimized Schwarz methods. SIAM Rev. 61:1 (2019), 3–76, 10.1137/16m109781x.
Dolean, V., Jolivet, P., Nataf, F., An Introduction to Domain Decomposition Methods: Algorithms, Theory and Parallel Implementation. 2015, Society for Industrial and Applied Mathematics, 10.1137/1.9781611974065.
Peng, Z., Lee, J.-F., Non-conformal domain decomposition method with mixed true second order transmission condition for solving large finite antenna arrays. IEEE Trans. Antennas Propag. 59:5 (2011), 1638–1651, 10.1109/TAP.2011.2123067.
Tournier, P.-H., Bonazzoli, M., Dolean, V., Rapetti, F., Hecht, F., Nataf, F., Aliferis, I., El Kanfoud, I., Migliaccio, C., de Buhan, M., Darbas, M., Semenov, S., Pichot, C., Numerical modeling and high-speed parallel computing: new perspectives on tomographic microwave imaging for brain stroke detection and monitoring. IEEE Antennas Propag. Mag. 59:5 (2017), 98–110, 10.1109/map.2017.2731199.
Marsic, N., Waltz, C., Lee, J.-F., Geuzaine, C., Domain decomposition methods for time-harmonic electromagnetic waves with high order Whitney forms. IEEE Trans. Magn. 52:3 (2016), 1–4, 10.1109/TMAG.2015.2476510.
Titchmarsh, E.C., The Theory of Functions. 2nd edition, 1976, Oxford University Press, London, England.
Royer, A., Béchet, E., Geuzaine, C., Gmsh-Fem: an efficient finite element library based on Gmsh. 14th WCCM-ECCOMAS Congress, 2021, 10.23967/wccm-eccomas.2020.161.
Saad, Y., Iterative Methods for Sparse Linear Systems. 2nd edition, 2003, Society for Industrial and Applied Mathematics, 10.1137/1.9780898718003.
Saad, Y., Schultz, M.H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Comput. 7:3 (1986), 856–869, 10.1137/0907058.
Balay, S., Gropp, W.D., McInnes, L.C., Smith, B.F., Efficient management of parallelism in object oriented numerical software libraries. Arge, E., Bruaset, A.M., Langtangen, H.P., (eds.) Modern Software Tools in Scientific Computing, 1997, Birkhäuser Press, 163–202, 10.1007/978-1-4612-1986-6_8.
Amestoy, P.R., Duff, I.S., Koster, J., L'Excellent, J.-Y., A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl. 23:1 (2001), 15–41, 10.1137/S0895479899358194.
Greenbaum, A., Pták, V., Strakoš, Z., Any nonincreasing convergence curve is possible for GMRES. SIAM J. Matrix Anal. Appl. 17:3 (1996), 465–469, 10.1137/s0895479894275030.
Haider, D.M., De Gersem, H., Golm, J., Koettig, T., Kurian, F., Marsic, N., Müller, W.F.O., Schmelz, M., Schwickert, M., Sieber, T., Stolz, R., Stöhlker, T., Tympel, V., Ucar, F., Zakosarenko, V., Versatile beamline cryostat for the cryogenic current comparator (CCC) for FAIR. Proceedings of the 8th International Beam Instrumentation Conference (IBIC'19), no. 8 in International Beam Instrumentation Conference, 2019, JACoW Publishing, Geneva, Switzerland, 78–81, 10.18429/JACoW-IBIC2019-MOPP007.
Seidel, P., Tympel, V., Neubert, R., Golm, J., Schmelz, M., Stolz, R., Zakosarenko, V., Sieber, T., Schwickert, M., Kurian, F., Schmidl, F., Stöhlker, T., Cryogenic current comparators for larger beamlines. IEEE Trans. Appl. Supercond. 28:4 (2018), 1–5, 10.1109/tasc.2018.2815647.
Boubendir, Y., Midura, D., Non-overlapping domain decomposition algorithm based on modified transmission conditions for the Helmholtz equation. Comput. Math. Appl. 75:6 (2018), 1900–1911, 10.1016/j.camwa.2017.07.027.
Similar publications
Sorry the service is unavailable at the moment. Please try again later.
This website uses cookies to improve user experience. Read more
Save & Close
Accept all
Decline all
Show detailsHide details
Cookie declaration
About cookies
Strictly necessary
Performance
Strictly necessary cookies allow core website functionality such as user login and account management. The website cannot be used properly without strictly necessary cookies.
This cookie is used by Cookie-Script.com service to remember visitor cookie consent preferences. It is necessary for Cookie-Script.com cookie banner to work properly.
Performance cookies are used to see how visitors use the website, eg. analytics cookies. Those cookies cannot be used to directly identify a certain visitor.
Used to store the attribution information, the referrer initially used to visit the website
Cookies are small text files that are placed on your computer by websites that you visit. Websites use cookies to help users navigate efficiently and perform certain functions. Cookies that are required for the website to operate properly are allowed to be set without your permission. All other cookies need to be approved before they can be set in the browser.
You can change your consent to cookie usage at any time on our Privacy Policy page.
