A numerical method for 3D time-dependent Maxwell’s equations in axisymmetric singular domains with arbitrary data
Abstract
In this article, we propose to solve the three-dimensional time-dependent Maxwell equations in a singular axisymmetric domain with arbitrary data. Due to the axisymmetric assumption, the singular computational domain boils down to a subset of R2. However, the electromagnetic field and other vector quantities still belong to R3. Taking advantage that the domain is transformed into a two-dimensional one, by doing Fourier analysis in the third dimension, one arrives to a sequence of singular problems set in a 2D domain. The mathematical tools of such problems have been exposed in [4,19]. Here, we derive a variational method from which we propose an original finite element numerical approach to solve the problem. Numerical experiments are also shown to illustrate that the method is able to capture the singular part of the solution. This approach can also be viewed as a generalization of the Singular Complement Method to three-dimensional problem.
Keyword : time-dependent Maxwell equations, Fourier analysis, singularities, axisymmetric geometry
How to Cite
Assous, F., & Raichik, I. (2023). A numerical method for 3D time-dependent Maxwell’s equations in axisymmetric singular domains with arbitrary data. Mathematical Modelling and Analysis, 28(3), 487–508. https://doi.org/10.3846/mma.2023.17553
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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F. Assous, P. Ciarlet, Jr. and S. Labrunie. Solution of axisymmetric Maxwell equations,. Math. Meth. Appl. Sci., 26(10):861–896, 2003. https://doi.org/10.1002/mma.400
F. Assous, P. Ciarlet, Jr. and S. Labrunie. Mathematical Foundations of Computational Electromagnetism. Appl. Math. Sc., AMS 198, Springer, 2018. https://doi.org/10.1007/978-3-319-70842-3
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(1):147–176, 2003. https://doi.org/10.1016/S0021-9991(03)00309-7
F. Assous, P. Ciarlet, Jr. and J. Segré. Numerical solution to the timedependent Maxwell equations in two-dimensional singular domain: The singular complement method. J. Comput. Phys., 161(1):218–249, 2000. https://doi.org/10.1006/jcph.2000.6499
F. Assous, P. Degond, P.A. Raviart and J. Segré. On a finite element method for solving the three-dimensional Maxwell equations. J. Comput. Phys., 109(2):222– 237, 1993. https://doi.org/10.1006/jcph.1993.1214
F. Assous, P. Degond and J. Segré. Numerical approximation of the Maxwell equations in inhomogeneous media by a p1 conforming finite element method. J. Comput. Phys., 128(2):363–380, 1996. https://doi.org/10.1006/jcph.1996.0217
F. Assous and I. Raichik. Solving numerically the static Maxwell equations in an axisymmetric singular geometry. Maths. Model. Anal., 20(1):9–29, 2015. https://doi.org/10.3846/13926292.2015.996615
F. Assous and I. Raichik. Numerical solution to the 3D static Maxwell equations in axisymmetric singular domains with arbitrary data. Comput. Meths. Applied Maths., 20:419–435, 2020. https://doi.org/10.1515/cmam-2018-0314
F. Ben Belgacem, C. Bernardi and F. Rapetti. Numerical analysis of a model for an axisymmetric guide for electromagnetic waves. I. the continuous problem and its Fourier expansion. Math. Methods Appl. Sci, 28(17):2007–2029, 2005. https://doi.org/10.1002/mma.649
F.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 Meth. App. Sci., 16(2):233–263, 2006. https://doi.org/10.1142/S0218202506001133
C. Bernardi and Y. Maday. Spectral methods for axisymmetric domains. Series in Applied Mathematics, Gauthier-Villars, Paris and North Holland, Amsterdam, 1999.
M.Sh. Birman and M.Z. Solomyak. L2-theory of the Maxwell operator in arbitrary domains. Russian Math. Surveys, 42(6):75–96, 1987. https://doi.org/10.1070/RM1987v042n06ABEH001505
M.Sh. Birman and M.Z. Solomyak. The Weyl asymptotic decomposition of the spectrum of the Maxwell operator for domain with lipschitzian boundary. Vestnik. Leningr. Univ. Math., 20:15–21, 1987.
S.C. Brenner, J. Gedicke and L.-Y. Sung. An adaptive P1 finite element method for two-dimensional Maxwell’s equations. Journal of Scientific Computing, 55:738–754, 2013. https://doi.org/10.1007/s10915-012-9658-8
C. Canuto, M.Y. Hussaini, A. Quarteroni and T. A. Zang. Spectral Methods in Fluid Dynamics. Springer-Verlag, New York, 1988. https://doi.org/10.1007/978-3-642-84108-8
Q. Chen and P. Monk. Introduction to applications of numerical analysis in time domain computational electromagnetism. Lecture Notes in Computational Science and Engineering, 2012. https://doi.org/10.1007/978-3-642-23914-4_3
P. Ciarlet, Jr., B. Jung, S. Kaddouri, S. Labrunie and J. Zou. The Fouriersingular complement method for Poisson equation. Part II: axisymmetric domains. Numer. Math., 102:583–610, 2006. https://doi.org/10.1007/s00211-005-0664-8
P. Ciarlet, Jr. and S. Labrunie. Numerical solution of Maxwell’s equations in axisymmetric domains with the Fourier singular complement method. Diff. Eq. & Applic., 3-1:113–155, 2011.
D.M. Copeland, J. Gopalakrishnan and J.E. Pasciak. A mixed method for axisymmetric div-curl systems. Math. Comp., 77:1941–1965, 2008. https://doi.org/10.1090/S0025-5718-08-02102-9
M. Costabel. A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains. Math. Meth. Appl. Sci., 12(4):365–368, 1990. https://doi.org/10.1002/mma.1670120406
M. Costabel, M. Dauge and S. Nicaise. Singularities of Maxwell interface problems. Model. Math. Anal. Num., 33(3):627–649, 1999. https://doi.org/10.1051/m2an:1999155
P. Grisvard. Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics. Pitman, London, 1985.
P. Grisvard. Singularities in boundary value problems. RMA 22, Masson, Paris, 1992.
F. Hecht. New development in FreeFem++. J. Numer. Math., 20(3-4):251–265, 2012. https://doi.org/10.1515/jnum-2012-0013
B. Heinrich. The Fourier-finite element method for Poisson’s equation in axisymmetric domains with edges. SIAM J. Numer. Anal., 33(5):1885–1911, 1996. https://doi.org/10.1137/S0036142994266108
B. Heinrich, S. Nicaise and B. Weber. Elliptic interface problems in axisymmetric domains II: Convergence analysis of the Fourier-finite element method. Adv. Math. Sci. Appl., 10:571–600, 2003.
J.S.Hestaven and T.Warburton. Nodal discontinuous Galerkin methods. Texts in Applied Mathematics, Springer, 2008. https://doi.org/10.1007/978-0-387-72067-8
J.L. Lions and E. Magenes. Non-Homogeneous Boundary Value Problems and Applications, vol. I. Springer, 1972. https://doi.org/10.1007/978-3-642-65161-8
B. Mercier and G. Raugel. Resolution d’un probleme aux limites dans un ouvert axisymétrique par élément finis en r,z et séries de Fourier en θ. R.A.I.R.O. Anal. numér., 16:405–461, 1982. https://doi.org/10.1051/m2an/1982160404051
J-C. Nédelec. Mixed finite elements in R3. Numer. Math., 35:315–341, 1980. https://doi.org/10.1007/BF01396415
J-C. Nédelec. A new family of mixed finite elements in R3. Numer. Math., 50:57–81, 1986. https://doi.org/10.1007/BF01389668
B. Nkemzi. Optimal convergence recovery for the Fourier-finite-element approximation of Maxwell’s equations in nonsmooth axisymmetric domains. Numer. Math., 57:989–1007, 2007. https://doi.org/10.1016/j.apnum.2006.09.006
C. Weber. A local compactness theorem for Maxwell’s equations. Math. Meth. Appl. Sci., 2(1):12–25, 1980. https://doi.org/10.1002/mma.1670020103
F. Assous, P. Ciarlet, Jr. and S. Labrunie. Solution of axisymmetric Maxwell equations,. Math. Meth. Appl. Sci., 26(10):861–896, 2003. https://doi.org/10.1002/mma.400
F. Assous, P. Ciarlet, Jr. and S. Labrunie. Mathematical Foundations of Computational Electromagnetism. Appl. Math. Sc., AMS 198, Springer, 2018. https://doi.org/10.1007/978-3-319-70842-3
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(1):147–176, 2003. https://doi.org/10.1016/S0021-9991(03)00309-7
F. Assous, P. Ciarlet, Jr. and J. Segré. Numerical solution to the timedependent Maxwell equations in two-dimensional singular domain: The singular complement method. J. Comput. Phys., 161(1):218–249, 2000. https://doi.org/10.1006/jcph.2000.6499
F. Assous, P. Degond, P.A. Raviart and J. Segré. On a finite element method for solving the three-dimensional Maxwell equations. J. Comput. Phys., 109(2):222– 237, 1993. https://doi.org/10.1006/jcph.1993.1214
F. Assous, P. Degond and J. Segré. Numerical approximation of the Maxwell equations in inhomogeneous media by a p1 conforming finite element method. J. Comput. Phys., 128(2):363–380, 1996. https://doi.org/10.1006/jcph.1996.0217
F. Assous and I. Raichik. Solving numerically the static Maxwell equations in an axisymmetric singular geometry. Maths. Model. Anal., 20(1):9–29, 2015. https://doi.org/10.3846/13926292.2015.996615
F. Assous and I. Raichik. Numerical solution to the 3D static Maxwell equations in axisymmetric singular domains with arbitrary data. Comput. Meths. Applied Maths., 20:419–435, 2020. https://doi.org/10.1515/cmam-2018-0314
F. Ben Belgacem, C. Bernardi and F. Rapetti. Numerical analysis of a model for an axisymmetric guide for electromagnetic waves. I. the continuous problem and its Fourier expansion. Math. Methods Appl. Sci, 28(17):2007–2029, 2005. https://doi.org/10.1002/mma.649
F.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 Meth. App. Sci., 16(2):233–263, 2006. https://doi.org/10.1142/S0218202506001133
C. Bernardi and Y. Maday. Spectral methods for axisymmetric domains. Series in Applied Mathematics, Gauthier-Villars, Paris and North Holland, Amsterdam, 1999.
M.Sh. Birman and M.Z. Solomyak. L2-theory of the Maxwell operator in arbitrary domains. Russian Math. Surveys, 42(6):75–96, 1987. https://doi.org/10.1070/RM1987v042n06ABEH001505
M.Sh. Birman and M.Z. Solomyak. The Weyl asymptotic decomposition of the spectrum of the Maxwell operator for domain with lipschitzian boundary. Vestnik. Leningr. Univ. Math., 20:15–21, 1987.
S.C. Brenner, J. Gedicke and L.-Y. Sung. An adaptive P1 finite element method for two-dimensional Maxwell’s equations. Journal of Scientific Computing, 55:738–754, 2013. https://doi.org/10.1007/s10915-012-9658-8
C. Canuto, M.Y. Hussaini, A. Quarteroni and T. A. Zang. Spectral Methods in Fluid Dynamics. Springer-Verlag, New York, 1988. https://doi.org/10.1007/978-3-642-84108-8
Q. Chen and P. Monk. Introduction to applications of numerical analysis in time domain computational electromagnetism. Lecture Notes in Computational Science and Engineering, 2012. https://doi.org/10.1007/978-3-642-23914-4_3
P. Ciarlet, Jr., B. Jung, S. Kaddouri, S. Labrunie and J. Zou. The Fouriersingular complement method for Poisson equation. Part II: axisymmetric domains. Numer. Math., 102:583–610, 2006. https://doi.org/10.1007/s00211-005-0664-8
P. Ciarlet, Jr. and S. Labrunie. Numerical solution of Maxwell’s equations in axisymmetric domains with the Fourier singular complement method. Diff. Eq. & Applic., 3-1:113–155, 2011.
D.M. Copeland, J. Gopalakrishnan and J.E. Pasciak. A mixed method for axisymmetric div-curl systems. Math. Comp., 77:1941–1965, 2008. https://doi.org/10.1090/S0025-5718-08-02102-9
M. Costabel. A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains. Math. Meth. Appl. Sci., 12(4):365–368, 1990. https://doi.org/10.1002/mma.1670120406
M. Costabel, M. Dauge and S. Nicaise. Singularities of Maxwell interface problems. Model. Math. Anal. Num., 33(3):627–649, 1999. https://doi.org/10.1051/m2an:1999155
P. Grisvard. Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics. Pitman, London, 1985.
P. Grisvard. Singularities in boundary value problems. RMA 22, Masson, Paris, 1992.
F. Hecht. New development in FreeFem++. J. Numer. Math., 20(3-4):251–265, 2012. https://doi.org/10.1515/jnum-2012-0013
B. Heinrich. The Fourier-finite element method for Poisson’s equation in axisymmetric domains with edges. SIAM J. Numer. Anal., 33(5):1885–1911, 1996. https://doi.org/10.1137/S0036142994266108
B. Heinrich, S. Nicaise and B. Weber. Elliptic interface problems in axisymmetric domains II: Convergence analysis of the Fourier-finite element method. Adv. Math. Sci. Appl., 10:571–600, 2003.
J.S.Hestaven and T.Warburton. Nodal discontinuous Galerkin methods. Texts in Applied Mathematics, Springer, 2008. https://doi.org/10.1007/978-0-387-72067-8
J.L. Lions and E. Magenes. Non-Homogeneous Boundary Value Problems and Applications, vol. I. Springer, 1972. https://doi.org/10.1007/978-3-642-65161-8
B. Mercier and G. Raugel. Resolution d’un probleme aux limites dans un ouvert axisymétrique par élément finis en r,z et séries de Fourier en θ. R.A.I.R.O. Anal. numér., 16:405–461, 1982. https://doi.org/10.1051/m2an/1982160404051
J-C. Nédelec. Mixed finite elements in R3. Numer. Math., 35:315–341, 1980. https://doi.org/10.1007/BF01396415
J-C. Nédelec. A new family of mixed finite elements in R3. Numer. Math., 50:57–81, 1986. https://doi.org/10.1007/BF01389668
B. Nkemzi. Optimal convergence recovery for the Fourier-finite-element approximation of Maxwell’s equations in nonsmooth axisymmetric domains. Numer. Math., 57:989–1007, 2007. https://doi.org/10.1016/j.apnum.2006.09.006
C. Weber. A local compactness theorem for Maxwell’s equations. Math. Meth. Appl. Sci., 2(1):12–25, 1980. https://doi.org/10.1002/mma.1670020103