Abstract
We propose a numerical method to solve the three-dimensional static Maxwell equations in a singular axisymmetric domain, generated by the rotation of a singular polygon around one of its sides. The mathematical tools and an in-depth study of the problem set in the meridian half-plane are exposed in [F. Assous, P. Ciarlet, Jr., S. Labrunie and J. Segré, Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: the singular complement method, J. Comput. Phys. 191 2003, 1, 147–176] and [P. Ciarlet, Jr. and S. Labrunie, Numerical solution of Maxwell’s equations in axisymmetric domains with the Fourier singular complement method, Differ. Equ. Appl. 3 2011, 1, 113–155]. Here, we derive a variational formulation and the corresponding approximation method. Numerical experiments are proposed, and show that the approach is able to capture the singular part of the solution. This article can also be viewed as a generalization of the Singular Complement Method to three-dimensional axisymmetric problems.
References
[1] F. Assous, P. Ciarlet, Jr. and S. Labrunie, Theoretical tools to solve the axisymmetric Maxwell equations, Math. Methods Appl. Sci. 25 (2002), no. 1, 49–78. 10.1002/mma.279Search in Google Scholar
[2] F. Assous, P. Ciarlet, Jr. and S. Labrunie, Solution of axisymmetric Maxwell equations, Math. Methods Appl. Sci. 26 (2003), no. 10, 861–896. 10.1002/mma.400Search in Google Scholar
[3] F. Assous, P. Ciarlet, Jr. and S. Labrunie, Mathematical Foundations of Computational Electromagnetism, Appl. Math. Sci. 198, Springer, Cham, 2018. 10.1007/978-3-319-70842-3Search in Google Scholar
[4] F. Assous, P. Ciarlet, Jr., S. Labrunie and J. Segré, Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: the singular complement method, J. Comput. Phys. 191 (2003), no. 1, 147–176. 10.1016/S0021-9991(03)00309-7Search in Google Scholar
[5] F. Assous, P. Ciarlet, Jr. and J. Segré, Numerical solution to the time-dependent Maxwell equations in two-dimensional singular domains: The singular complement method, J. Comput. Phys. 161 (2000), no. 1, 218–249. 10.1006/jcph.2000.6499Search in Google Scholar
[6] F. Assous, P. Degond, E. Heintze, P.-A. Raviart and J. Segre, On a finite-element method for solving the three-dimensional Maxwell equations, J. Comput. Phys. 109 (1993), no. 2, 222–237. 10.1006/jcph.1993.1214Search in Google Scholar
[7] F. Assous and I. Raichik, Solving numerically the static Maxwell equations in an axisymmetric singular geometry, Math. Model. Anal. 20 (2015), no. 1, 9–29. 10.3846/13926292.2015.996615Search in Google Scholar
[8] R. Becker, P. Hansbo and R. Stenberg, A finite element method for domain decomposition with non-matching grids, M2AN Math. Model. Numer. Anal. 37 (2003), no. 2, 209–225. 10.1051/m2an:2003023Search in Google Scholar
[9] Z. Belhachmi, C. Bernardi, S. Deparis and F. Hecht, A truncated Fourier/finite element discretization of the Stokes equations in an axisymmetric domain, Math. Models Methods Appl. Sci. 16 (2006), no. 2, 233–263. 10.1142/S0218202506001133Search in Google Scholar
[10] C. Bernardi, M. Dauge and Y. Maday, Spectral Methods for Axisymmetric Domains, Ser. Appl. Math. (Paris) 3, Gauthier-Villars, Paris, 1999. Search in Google Scholar
[11]
M. S. Birman and M. Z. Solomyak,
[12] M. S. Birman and M. Z. Solomyak, Weyl asymptotics of the spectrum of the Maxwell operator for domains with a Lipschitz boundary, Vestnik Leningrad. Univ. Math. 20 (1987), 15–21. Search in Google Scholar
[13]
S. C. Brenner, J. Gedicke and L.-Y. Sung,
An adaptive
[14] C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Seri. Comput. Phys., Springer, New York, 1988. 10.1007/978-3-642-84108-8Search in Google Scholar
[15] Q. Chen and P. Monk, Introduction to applications of numerical analysis in time domain computational electromagnetism, Frontiers in Numerical Analysis (Durham 2010), Lect. Notes Comput. Sci. Eng. 85, Springer, Heidelberg (2012), 149–225. 10.1007/978-3-642-23914-4_3Search in Google Scholar
[16] P. Ciarlet, Jr., B. Jung, S. Kaddouri, S. Labrunie and J. Zou, The Fourier singular complement method for the Poisson problem. I. Prismatic domains, Numer. Math. 101 (2005), no. 3, 423–450. 10.1007/s00211-005-0621-6Search in Google Scholar
[17] P. Ciarlet, Jr., B. Jung, S. Kaddouri, S. Labrunie and J. Zou, The Fourier singular complement method for the Poisson problem. II. Axisymmetric domains, Numer. Math. 102 (2006), no. 4, 583–610. 10.1007/s00211-005-0664-8Search in Google Scholar
[18] P. Ciarlet, Jr. and S. Labrunie, Numerical solution of Maxwell’s equations in axisymmetric domains with the Fourier singular complement method, Differ. Equ. Appl. 3 (2011), no. 1, 113–155. 10.7153/dea-03-08Search in Google Scholar
[19] P. Ciarlet, Jr. and J. Zou, Finite element convergence for the Darwin model to Maxwell’s equations, RAIRO Modél. Math. Anal. Numér. 31 (1997), no. 2, 213–249. 10.1051/m2an/1997310202131Search in Google Scholar
[20] D. M. Copeland, J. Gopalakrishnan and J. E. Pasciak, A mixed method for axisymmetric div-curl systems, Math. Comp. 77 (2008), no. 264, 1941–1965. 10.1090/S0025-5718-08-02102-9Search in Google Scholar
[21] M. Costabel, A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains, Math. Methods Appl. Sci. 12 (1990), no. 4, 365–368. 10.1002/mma.1670120406Search in Google Scholar
[22] M. Costabel and M. Dauge, Singularities of electromagnetic fields in polyhedral domains, Arch. Ration. Mech. Anal. 151 (2000), no. 3, 221–276. 10.1007/s002050050197Search in Google Scholar
[23] V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms, Springer Ser. Comput. Math. 5, Springer, Berlin, 1986. 10.1007/978-3-642-61623-5Search in Google Scholar
[24] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math. 24, Pitman, Boston, 1985. Search in Google Scholar
[25] P. Grisvard, Singularities in Boundary Value Problems, Rech. Math. Appl. 22, Masson, Paris, 1992. Search in Google Scholar
[26] F. Hecht, New development in freefem++, J. Numer. Math. 20 (2012), no. 3–4, 251–265. 10.1515/jnum-2012-0013Search in Google Scholar
[27] B. Heinrich, The Fourier-finite-element method for Poisson’s equation in axisymmetric domains with edges, SIAM J. Numer. Anal. 33 (1996), no. 5, 1885–1911. 10.1137/S0036142994266108Search in Google Scholar
[28] B. Heinrich, S. Nicaise and B. Weber, Elliptic interface problems in axisymmetric domains. II. The Fourier-finite-element approximation of non-tensorial singularities, Adv. Math. Sci. Appl. 10 (2000), no. 2, 571–600. Search in Google Scholar
[29] J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods. Algorithms, Analysis, and Applications, Texts Appl. Math. 54, Springer, New York, 2008. 10.1007/978-0-387-72067-8Search in Google Scholar
[30] Y. P. Kim and J. R. Kweon, The Fourier-finite element method for the Poisson problem on a non-convex polyhedral cylinder, J. Comput. Appl. Math. 233 (2009), no. 4, 951–968. 10.1016/j.cam.2009.08.097Search in Google Scholar
[31] B. Mercier and G. Raugel, Résolution d’un problème aux limites dans un ouvert axisymétrique par éléments finis en r, z et séries de Fourier en θ, RAIRO Anal. Numér. 16 (1982), no. 4, 405–461. 10.1051/m2an/1982160404051Search in Google Scholar
[32] P. Monk, Finite Element Methods for Maxwell’s Equations, Numer. Math. Sci. Comput., Oxford University, New York, 2003. 10.1093/acprof:oso/9780198508885.001.0001Search in Google Scholar
[33]
J.-C. Nédélec,
Mixed finite elements in
[34]
J.-C. Nédélec,
A new family of mixed finite elements in
[35] B. Nkemzi, Optimal convergence recovery for the Fourier-finite-element approximation of Maxwell’s equations in nonsmooth axisymmetric domains, Appl. Numer. Math. 57 (2007), no. 9, 989–1007. 10.1016/j.apnum.2006.09.006Search in Google Scholar
[36] C. Weber, A local compactness theorem for Maxwell’s equations, Math. Methods Appl. Sci. 2 (1980), no. 1, 12–25. 10.1002/mma.1670020103Search in Google Scholar
[37]
I. Yousept,
Optimal control of quasilinear
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