A defect-correction nodal finite element method for time-dependent Maxwell's equations on polygonal domains
DOI:
https://doi.org/10.33993/jnaat551-1664Keywords:
Maxwell's Equations, Crank-Nicolson Method, Singular functionAbstract
This paper develops a nodal finite element Crank-Nicolson method of lines to solve the time-dependent Maxwell's equations on polygonal domains with reentrant corners. Nodal Finite Element methods are used to solve Maxwell's equations with an optimal convergence rate when the domain is convex or has a smooth boundary, but may fail to converge if the domain has a reentrant corner. The Defect-Correction method presented is based on a decomposition of the solution in terms of Fourier and Bessel's series, an extraction of the singular function and an approximation of the regular part of the solution. Optimal convergence results are recovered using the method in both the energy norm or the $L^2$-norm.
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[1] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer Science & Business Media, New York (2008). DOI: https://doi.org/10.1007/978-0-387-75934-0
[2] S.C. Brenner and L-Y. Sung, Multigrid method for the computation of singular solutions and stress intensity factors II: crack singularities, BIT Numerical Mathematics, 37 (1997) no. 3, pp. 623–643. DOI: https://doi.org/10.1007/BF02510243
[3] P. Ciarlet, The Finite Element Method for Elliptic Problems, Society for Industrial and Applied Mathematics, Amsterdam (2002). DOI: https://doi.org/10.1137/1.9780898719208
[4] P. Ciarlet, Augmented formulations for solving Maxwell equations, Computer Methods in Applied Mechanics and Engineering, 194 (2005) no. 2-5, pp. 559–586. DOI: https://doi.org/10.1016/j.cma.2004.05.021
[5] M. Costabel and M. Dauge, Singularities of electromagnetic fields in polyhedral domains, Archives of Rational Mechanics and Analysis, 151 (2000) no. 3, pp. 221–276. DOI: https://doi.org/10.1007/s002050050197
[6] M. Costabel, M. Dauge and S. Nicaise, Singularities of Maxwell interface problems, Mathematical Modelling and Numerical Analysis, 33 (1999) no. 3, pp. 627–649. DOI: https://doi.org/10.1051/m2an:1999155
[7] M. Dauge, Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions, Lecture Notes in Mathematics, Springer-Verlag, Berlin (1988). DOI: https://doi.org/10.1007/BFb0086682
[8] P. Grisvard, Singularities in Boundary Value Problems, Recherches en mathématiques appliquées, Masson (1992).
[9] B. Nkemzi, On the solution of Maxwell's equations in polygonal domains, Mathematical Methods in Applied Science, 29 (2006) no. 9, pp. 1053–1080. DOI: https://doi.org/10.1002/mma.717
[10] B. Nkemzi, On singularities of solution of Maxwell's equations in axisymmetric domains with conical points, Mathematical Methods in Applied Science, 30 (2007) no. 8, pp. 877–888. DOI: https://doi.org/10.1002/mma.812
[11] B. Nkemzi, On the Coefficients of the Singularities of the Solution of Maxwell's Equations near Polyhedral Edges, Mathematical Problems in Engineering, 2016 (2016), pp. 1–17. DOI: https://doi.org/10.1155/2016/7965642
[12] B. Nkemzi and J. Nkeck, A predictor-corrector finite element method for Maxwell's equations in polygonal domains, Mathematical Problems in Engineering, 2020 (2020), pp. 1–13. DOI: https://doi.org/10.1155/2020/3502513
[13] M. Costabel and M. Dauge, Weighted regularization of Maxwell equations in polyhedral domains. A rehabilitation of Nodal finite elements, Numerische Mathematik, 93 (2002) no. 2, pp. 239–277. DOI: https://doi.org/10.1007/s002110100388
[14] M. Costabel, A coercive bilinear form for Maxwell's equations, Journal of Mathematical Analysis and Applications, 157 (1991) no. 2, pp. 527–541. DOI: https://doi.org/10.1016/0022-247X(91)90104-8
[15] C. Hazard and M. Lenoir, On the solution of time-harmonic scattering problems for Maxwell's equations, SIAM Journal of Mathematical Analysis, 27 (1996) no. 6, pp. 1597–1630. DOI: https://doi.org/10.1137/S0036141094271259
[16] A.S. Bonnet-Ben Dhia, C. Hazard and S. Lorenghel, A Singular Field Method for the Solution of Maxwell's Equations in Polyhedral Domains, SIAM Journal of Applied Mathematics, 59 (1999) no. 6, pp. 2028–2044. DOI: https://doi.org/10.1137/S0036139997323383
[17] C. Hazard and S. Lorenghel, A Singular field method for the solution of Maxwell's equations: numerical aspects for 2D magnetostatics, SIAM Journal of Applied Mathematics, 40 (2002) no. 3, pp. 1021–1040. DOI: https://doi.org/10.1137/S0036142900375761
[18] E. Jamelot, Elements finis nodaux pour les Equations de Maxwell, Comptes Rendus de l'Académie des Sciences de Paris, 339 (2004) no. 11, pp. 809–814. DOI: https://doi.org/10.1016/j.crma.2004.10.020
[19] A. Bonito, J. Guermond and F. Luddens, An Interior Penalty Method with C0 Finite Elements for the Approximation of the Maxwell Equations in Heterogeneous Media: Convergence Analysis with Minimal Regularity, ESAIM: Mathematical Modelling and Numerical Analysis, 50 (2016) no. 5, pp. 1457–1489. DOI: https://doi.org/10.1051/m2an/2015086
[20] A. Buffa, C. Ciarlet Jr and E. Jamelot, Solving electromagnetic eigenvalue problems in polyhedral domains with nodal finite elements, Numerische Mathematik, 113 (2009) no. 4, pp. 497–518. DOI: https://doi.org/10.1007/s00211-009-0246-2
[21] R. Otin, Regularized Maxwell equations and nodal finite elements for electromagnetic field computations in frequency domain, Universitat Politècnica de Catalunya (2011).
[22] R. Otin, Regularized Maxwell Equations and Nodal Finite Elements for Electromagnetic Field Computations, Electromagnetics, 30 (2010) no. 1-2, pp. 190–204. DOI: https://doi.org/10.1080/02726340903485489
[23] R. Codina, A stabilized finite element method for generalized stationary incompressible flows, Computer Methods in Applied Mechanics and Engineering, 190 (2001) no. 20-21, pp. 2681–2706. DOI: https://doi.org/10.1016/S0045-7825(00)00260-7
[24] R. Codina, Analysis of a stabilized finite element approximation of the Oseen equations using orthogonal subscales, Applied Numerical Mathematics, 58 (2008) no. 3, pp. 264–283. DOI: https://doi.org/10.1016/j.apnum.2006.11.011
[25] R. Codina and S. Badia, A nodal-based finite element approximation of the Maxwell problem suitable for singular solutions, SIAM J. Numer. Anal., 50 (2012) no. 2, pp. 398–417. DOI: https://doi.org/10.1137/110835360
[26] H.P. Duan, P. Lin, P. Saikrisman and R.C.E. Tan, A least-squares finite element method for the magnetostatic problem in a multiply connected Lipschitz domain, SIAM Journal on Numerical Analysis, 45 (2007) no. 6, pp. 2537–2563. DOI: https://doi.org/10.1137/050640102
[27] H.P. Duan, F. Jia, P. Lin and R.C.E. Tan, The local L²-projected C0 finite element method for Maxwell Problem, SIAM Journal on Numerical Analysis, 47 (2009) no. 2, pp. 1274–1303. DOI: https://doi.org/10.1137/070707749
[28] A. Bonito and J. Guermond, Approximation of the eigenvalue problem for the time-harmonic Maxwell system by continuous Lagrange finite elements, Mathematics of Computation, 80 (2011) no. 276, pp. 1887–1910. DOI: https://doi.org/10.1090/S0025-5718-2011-02464-6
[29] J.L. Nkeck, A Full Predictor-Corrector Finite Element Method for the One-Dimensional Heat Equation with Time-Dependent Singularities, Journal of Applied Mathematics and Physics, 12 (2024) no. 4, pp. 1364–1382. DOI: https://doi.org/10.4236/jamp.2024.124084
[30] B. Nkemzi and J. Nkeck, A predictor-corrector finite element method for parabolic problems on polygonal domains, Numerical Algorithms (2025) (in press). DOI: https://doi.org/10.1007/s11075-025-02243-1
[31] J-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer Berlin Heidelberg (1972). DOI: https://doi.org/10.1007/978-3-642-65217-2
[32] G. Baker, Error Estimates for Finite Element Methods for Second Order Hyperbolic Equations, SIAM J. Numer. Anal., 13 (1976) no. 4, pp. 564–576. DOI: https://doi.org/10.1137/0713048
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