Mortar spectral method in axisymmetric domains
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 1, p. 33-55

We consider the Laplace equation posed in a three-dimensional axisymmetric domain. We reduce the original problem by a Fourier expansion in the angular variable to a countable family of two-dimensional problems. We decompose the meridian domain, assumed polygonal, in a finite number of rectangles and we discretize by a spectral method. Then we describe the main features of the mortar method and use the algorithm Strang Fix to improve the accuracy of our discretization.

DOI : https://doi.org/10.1051/m2an/2012018
Classification:  65N35,  65N55
Keywords: axisymmetric domains, mortar method, spectral methods, Laplace equation
@article{M2AN_2013__47_1_33_0,
     author = {Mani Aouadi, Saloua and Satouri, Jamil},
     title = {Mortar spectral method in axisymmetric domains},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {1},
     year = {2013},
     pages = {33-55},
     doi = {10.1051/m2an/2012018},
     zbl = {1277.65101},
     mrnumber = {2968694},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2013__47_1_33_0}
}
Mani Aouadi, Saloua; Satouri, Jamil. Mortar spectral method in axisymmetric domains. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 1, pp. 33-55. doi : 10.1051/m2an/2012018. http://www.numdam.org/item/M2AN_2013__47_1_33_0/

[1] A.B. Abdallah, F.B. Belgacem, Y. Maday and F. Rapetti, Mortaring the two-dimensional edge finite elements for the discretization of some electromagnetic models. Math. Mod. Methods Appl. Sci. 14 (2004) 1635-1656. | MR 2103094 | Zbl 1100.78017

[2] M. Azaïez, C. Bernardi, M. Dauge and Y. Maday, Spectral Methods for Axisymetric Domains. Series in Appl. Math. 3 (1999). | Zbl 0929.35001

[3] F.B. Belgacem, C. Bernardi and F. Rapetti, Numerical analysis of a model for an axisymmetric guide for electromagnetic waves. Part I : The continuous problem and its Fourier expansion. Math. Meth. Appl. Sci. 28 (2005) 2007-2029. | MR 2176904 | Zbl 1125.35099

[4] C. Bernardi and Y. Maday, Properties of some weighted Sobolev spaces and application to spectral approximations. SIAM J. Numer. Anal. 26 (1989) 769-829. | MR 1005511 | Zbl 0675.65114

[5] C. Bernardi and Y. Maday, Approximations spectrales de problèmes aux limites elliptiques. Math. Appl. 10 (1992). | Zbl 0773.47032

[6] C. Bernardi, M. Dauge and M. Azaïez, Numerical Analysis and Spectral Methods in Axisymetric Problems. Rapport Interne, Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie (1995).

[7] S. Bertoluzza, S. Falletta and V. Perrier, The Mortar method in the wavelet context. Model. Math. Anal. Numer. 35 (2001) 647-673. | Numdam | MR 1862873 | Zbl 0995.65131

[8] H. Brezis, Analyse fonctionnelle, in Théorie et Applications. Masson, Paris (1983). | MR 697382 | Zbl 1147.46300

[9] N. Chorfi, Traitement de singularités géométriques par méthode d'éléments spectraux avec joints. Thèse de l'Université Pierre et Marie Curie, Paris VI (1998).

[10] M. Dauge, Elliptic Boundary Value Problems on Corner Domains. Lect. Notes Math. 1341 (1988). | MR 961439 | Zbl 0668.35001

[11] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, in Theory and Algorithms. Springer-Verlag (1986). | MR 851383 | Zbl 0585.65077

[12] P. Grisvard, Singularities in boundary value problems, in Collect. RMA 22 (1992). | MR 1173209 | Zbl 0766.35001

[13] P. Le Tallec, Domain decomposition methods in computational mechanics, in Comput. Mech. Adv. North-Holland (1994). | MR 1263805 | Zbl 0802.73079

[14] R. Pasquetti, L.F. Pavarino, F. Rapetti and E. Zampieri, Overlapping Schwarz methods for Fekete and Gauss-Lobatto spectral elements. SIAM J. Scient. Comput. 29 (2007) 1073-1092. | MR 2318698 | Zbl 1141.65394

[15] Y. Maday, C. Mavriplis and A.T. Patera, Nonconforming mortar element methods : application to spectral discretizations, in Domain decomposition methods. SIAM (1989) 392-418. | MR 992030 | Zbl 0692.65055

[16] J. Satouri, Méthodes d'éléments spectraux avec joints pour des géométries axisymétriques. Thèse de l'Université Pierre et Marie Curie, Paris VI (2010).

[17] G. Strang and G.J. Fix, An Analysis of the Finite Element Method, in Automatic Computation. Prentice Hall Serie (1973). | MR 443377 | Zbl 0356.65096