Numerical Solution to the 3D Static Maxwell Equations in Axisymmetric Singular Domains with Arbitrary Data

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.279Suche 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.400Suche in Google Scholar

[3] F. Assous, P. Ciarlet, Jr. and S. Labrunie, Mathematical Foundations of Computational Electromagnetism, Appl. Math. Sci. 198, Springer, Cham, 2018. 10.1007/978-3-319-70842-3Suche 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)00309-7Suche 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.2000.6499Suche 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.1993.1214Suche 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. 10.3846/13926292.2015.996615Suche 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:2003023Suche 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/S0218202506001133Suche 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. Suche in Google Scholar

[11] M. S. Birman and M. Z. Solomyak, L 2 -theory of the Maxwell operator in arbitrary domains, Russian Math. Surveys 42 (1987), 75–96. 10.1070/RM1987v042n06ABEH001505Suche in Google Scholar

[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. Suche in Google Scholar

[13] S. C. Brenner, J. Gedicke and L.-Y. Sung, An adaptive P 1 finite element method for two-dimensional Maxwell’s equations, J. Sci. Comput. 55 (2013), no. 3, 738–754. 10.1007/s10915-012-9658-8Suche in Google Scholar

[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. 10.1007/978-3-642-84108-8Suche 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_3Suche 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/s00211-005-0621-6Suche 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/s00211-005-0664-8Suche 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-03-08Suche 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/1997310202131Suche 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/S0025-5718-08-02102-9Suche 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.1670120406Suche 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/s002050050197Suche 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. 10.1007/978-3-642-61623-5Suche in Google Scholar

[24] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math. 24, Pitman, Boston, 1985. Suche in Google Scholar

[25] P. Grisvard, Singularities in Boundary Value Problems, Rech. Math. Appl. 22, Masson, Paris, 1992. Suche in Google Scholar

[26] F. Hecht, New development in freefem++, J. Numer. Math. 20 (2012), no. 3–4, 251–265. 10.1515/jnum-2012-0013Suche 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/S0036142994266108Suche 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. Suche 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. 10.1007/978-0-387-72067-8Suche 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.2009.08.097Suche 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/1982160404051Suche 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/9780198508885.001.0001Suche in Google Scholar

[33] J.-C. Nédélec, Mixed finite elements in 𝐑 3 , Numer. Math. 35 (1980), no. 3, 315–341. 10.1007/BF01396415Suche in Google Scholar

[34] J.-C. Nédélec, A new family of mixed finite elements in 𝐑 3 , Numer. Math. 50 (1986), no. 1, 57–81. 10.1007/BF01389668Suche in Google Scholar

[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.2006.09.006Suche 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.1670020103Suche in Google Scholar

[37] I. Yousept, Optimal control of quasilinear H ⁢ ( curl ) -elliptic partial differential equations in magnetostatic field problems, SIAM J. Control Optim. 51 (2013), no. 5, 3624–3651. 10.1137/120904299Suche in Google Scholar