An unfitted finite element method with direct extension stabilization for time-harmonic Maxwell problems on smooth domains. (English) Zbl 1542.65167
Summary: We propose an unfitted finite element method for numerically solving the time-harmonic Maxwell equations on a smooth domain. The embedded boundary of the domain is allowed to cut through the background mesh arbitrarily. The unfitted scheme is based on a mixed interior penalty formulation, where the Nitsche penalty method is applied to enforce the boundary condition in a weak sense, and a penalty stabilization technique is adopted based on a local direct extension operator to ensure the stability for cut elements. We prove the inf-sup stability and obtain optimal convergence rates under the energy norm and the \(L^2\) norm for both variables. Numerical examples in both two and three dimensions are presented to illustrate the accuracy of the method.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 78A25 | Electromagnetic theory (general) |
| 78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |
| 35B35 | Stability in context of PDEs |
| 35Q61 | Maxwell equations |
Software:
CutFEMReferences:
| [1] | Adams, R.A., Fournier, J.J.F. : Sobolev Spaces, second ed., Pure and Applied Mathematics (Amsterdam), vol. 140. Elsevier/Academic Press, Amsterdam (2003) · Zbl 1098.46001 |
| [2] | Amrouche, C.; Bernardi, C.; Dauge, M.; Girault, V., Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci., 21, 9, 823-864, 1998 · Zbl 0914.35094 · doi:10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B |
| [3] | Anselmann, M.; Bause, M., Cut finite element methods and ghost stabilization techniques for space-time discretizations of the Navier-Stokes equations, Internat. J. Numer. Methods Fluids, 94, 7, 775-802, 2022 · doi:10.1002/fld.5074 |
| [4] | Babuška, I.; Banerjee, U., Stable generalized finite element method (SGFEM), Comput. Methods Appl. Mech. Engrg., 201, 204, 91-111, 2012 · Zbl 1239.74093 · doi:10.1016/j.cma.2011.09.012 |
| [5] | Badia, S.; Verdugo, F.; Martín, A., The aggregated unfitted finite element method for elliptic problems, Comput. Methods Appl. Mech. Engrg., 336, 533-553, 2018 · Zbl 1440.65175 · doi:10.1016/j.cma.2018.03.022 |
| [6] | Bermúdez, A.; Rodríguez, R.; Salgado, P., A finite element method with Lagrange multipliers for low-frequency harmonic Maxwell equations, SIAM J. Numer. Anal., 40, 5, 1823-1849, 2002 · Zbl 1033.78009 · doi:10.1137/S0036142901390780 |
| [7] | Bordas, S.P.A., Burman, E., Larson, M.G., Olshanskii, M.A. (eds.) : Geometrically unfitted finite element methods and applications. Lecture Notes in Computational Science and Engineering, vol. 121, Springer, Cham, 2017. (2016) · Zbl 1392.65006 |
| [8] | Brenner, SC; Li, F.; Sung, L-Y, A locally divergence-free nonconforming finite element method for the time-harmonic Maxwell equations, Math. Comp., 76, 258, 573-595, 2007 · Zbl 1126.78017 · doi:10.1090/S0025-5718-06-01950-8 |
| [9] | Brenner, S.C., Scott, L.R. : The Mathematical Theory of Finite Element Methods, third ed., Texts in Applied Mathematics, vol. 15, Springer, New York, (2008) · Zbl 1135.65042 |
| [10] | Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, 1991, New York: Springer-Verlag, New York · Zbl 0788.73002 · doi:10.1007/978-1-4612-3172-1 |
| [11] | Burman, E., Ghost penalty, C. R. Math. Acad. Sci. Paris, 348, 21-22, 1217-1220, 2010 · Zbl 1204.65142 · doi:10.1016/j.crma.2010.10.006 |
| [12] | Burman, E.; Cicuttin, M.; Delay, G.; Ern, A., An unfitted hybrid high-order method with cell agglomeration for elliptic interface problems, SIAM J. Sci. Comput., 43, 2, A859-A882, 2021 · Zbl 1475.65186 · doi:10.1137/19M1285901 |
| [13] | Burman, E.; Claus, S.; Hansbo, P.; Larson, MG; Massing, A., CutFEM: discretizing geometry and partial differential equations, Internat. J. Numer. Methods Engrg., 104, 7, 472-501, 2015 · Zbl 1352.65604 · doi:10.1002/nme.4823 |
| [14] | Cao, S.; Chen, L.; Guo, R., A virtual finite element method for two-dimensional Maxwell interface problems with a background unfitted mesh, Math. Models Methods Appl. Sci., 31, 14, 2907-2936, 2021 · Zbl 1478.65117 · doi:10.1142/S0218202521500652 |
| [15] | R. Casagrande, R. Hiptmair, and J. Ostrowski, An a priori error estimate for interior penalty discretizations of the curl-curl operator on non-conforming meshes, J. Math. Ind. 6 (2016), Art. 4, 25 |
| [16] | Chen, Z.; Du, Q.; Zou, J., Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients, SIAM J. Numer. Anal., 37, 5, 1542-1570, 2000 · Zbl 0964.78017 · doi:10.1137/S0036142998349977 |
| [17] | Chen, Z.; Wang, L.; Zheng, W., An adaptive multilevel method for time-harmonic Maxwell equations with singularities, SIAM J. Sci. Comput., 29, 1, 118-138, 2007 · Zbl 1136.78013 · doi:10.1137/050636012 |
| [18] | Chen, Z.; Xiao, Y.; Zhang, L., The adaptive immersed interface finite element method for elliptic and Maxwell interface problems, J. Comput. Phys., 228, 14, 5000-5019, 2009 · Zbl 1172.78008 · doi:10.1016/j.jcp.2009.03.044 |
| [19] | Chen, Z.; Zou, J., Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79, 2, 175-202, 1998 · Zbl 0909.65085 · doi:10.1007/s002110050336 |
| [20] | Costabel, M.; Dauge, M., Singularities of electromagnetic fields in polyhedral domains, Arch. Ration. Mech. Anal., 151, 3, 221-276, 2000 · Zbl 0968.35113 · doi:10.1007/s002050050197 |
| [21] | Cui, T.; Leng, W.; Liu, H.; Zhang, L.; Zheng, W., High-order numerical quadratures in a tetrahedron with an implicitly defined curved interface, ACM Trans. Math. Software, 46, 1, 18, 2020 · Zbl 1484.65046 · doi:10.1145/3372144 |
| [22] | Demmel, JW; Eisenstatm, SC; Gilbert, JR; Li, XS; Liu, JWH, A supernodal approach to sparse partial pivoting, SIAM J. Matrix Anal. Appl., 20, 3, 720-755, 1999 · Zbl 0931.65022 · doi:10.1137/S0895479895291765 |
| [23] | Duan, H.; Tan, RCE; Yang, S-Y; You, C-S, A mixed \(H^1\)-conforming finite element method for solving Maxwell’s equations with non-\(H^1\) solution, SIAM J. Sci. Comput., 40, 1, A224-A250, 2018 · Zbl 1383.78037 · doi:10.1137/16M1078082 |
| [24] | Ern, A.; Guermond, J-L, Analysis of the edge finite element approximation of the Maxwell equations with low regularity solutions, Comput. Math. Appl., 75, 3, 918-932, 2018 · Zbl 1409.65068 · doi:10.1016/j.camwa.2017.10.017 |
| [25] | Feng, X.; Wu, H., An absolutely stable discontinuous Galerkin method for the indefinite time-harmonic Maxwell equations with large wave number, SIAM J. Numer. Anal., 52, 5, 2356-2380, 2014 · Zbl 1315.78011 · doi:10.1137/120902112 |
| [26] | Fries, T-P; Belytschko, T., The extended/generalized finite element method: an overview of the method and its applications, Internat. J. Numer. Methods Engrg., 84, 3, 253-304, 2010 · Zbl 1202.74169 · doi:10.1002/nme.2914 |
| [27] | Galdi, G.P. : An introduction to the mathematical theory of the Navier-Stokes equations, second ed., Springer Monographs in Mathematics, Springer, New York, 2011, Steady-state problems · Zbl 1245.35002 |
| [28] | Girault, V.; Raviart, P-A, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, 1986, Berlin: Springer-Verlag, Berlin · Zbl 0585.65077 · doi:10.1007/978-3-642-61623-5 |
| [29] | Grote, MJ; Schneebeli, A.; Schötzau, D., Interior penalty discontinuous Galerkin method for Maxwell’s equations: energy norm error estimates, J. Comput. Appl. Math., 204, 2, 375-386, 2007 · Zbl 1136.78014 · doi:10.1016/j.cam.2006.01.044 |
| [30] | Gürkan, C.; Massing, A., A stabilized cut discontinuous Galerkin framework for elliptic boundary value and interface problems, Comput. Methods Appl. Mech. Engrg., 348, 466-499, 2019 · Zbl 1440.65208 · doi:10.1016/j.cma.2018.12.041 |
| [31] | Gürkan, C.; Massing, A., A stabilized cut discontinuous Galerkin framework for elliptic boundary value and interface problems, Comput. Methods Appl. Mech. Engrg., 348, 466-499, 2019 · Zbl 1440.65208 · doi:10.1016/j.cma.2018.12.041 |
| [32] | Guzmán, J.; Olshanskii, M., Inf-sup stability of geometrically unfitted Stokes finite elements, Math. Comp., 87, 313, 2091-2112, 2018 · Zbl 1391.76121 · doi:10.1090/mcom/3288 |
| [33] | Han, Y., Chen, H., Wang, X., Xie, X.: EXtended HDG methods for second order elliptic interface problems. J. Sci. Comput. 84(1), 29 (2020). (Paper No. 22) · Zbl 1452.65339 |
| [34] | Han, Y., Wang, X.-P., Xie, X. : An interface/boundary-unfitted eXtended HDG method for linear elasticity problems. J. Sci. Comput. 94 (2023). (Paper No. 61) · Zbl 07698825 |
| [35] | Hansbo, A.; Hansbo, P., An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, Comput. Methods Appl. Mech. Engrg., 191, 47-48, 5537-5552, 2002 · Zbl 1035.65125 · doi:10.1016/S0045-7825(02)00524-8 |
| [36] | Hansbo, P.; Larson, MG; Zahedi, S., A cut finite element method for a Stokes interface problem, Appl. Numer. Math., 85, 90-114, 2014 · Zbl 1299.76136 · doi:10.1016/j.apnum.2014.06.009 |
| [37] | Hiptmair, R., Finite elements in computational electromagnetism, Acta Numer, 11, 237-339, 2002 · Zbl 1123.78320 · doi:10.1017/S0962492902000041 |
| [38] | Houston, P.; Perugia, I.; Schneebeli, D.; Schötzau, A., Interior penalty method for the indefinite time-harmonic Maxwell equations, Numer. Math., 100, 3, 485-518, 2005 · Zbl 1071.65155 · doi:10.1007/s00211-005-0604-7 |
| [39] | Houston, P.; Perugia, I.; Schötzau, D., Mixed discontinuous Galerkin approximation of the Maxwell operator, SIAM J. Numer. Anal., 42, 1, 434-459, 2004 · Zbl 1084.65115 · doi:10.1137/S003614290241790X |
| [40] | Huang, P.; Wu, H.; Xiao, Y., An unfitted interface penalty finite element method for elliptic interface problems, Comput. Methods Appl. Mech. Engrg., 323, 439-460, 2017 · Zbl 1439.74422 · doi:10.1016/j.cma.2017.06.004 |
| [41] | Johansson, A.; Larson, MG, A high order discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary, Numer. Math., 123, 4, 607-628, 2013 · Zbl 1269.65126 · doi:10.1007/s00211-012-0497-1 |
| [42] | Lehrenfeld, C., High order unfitted finite element methods on level set domains using isoparametric mappings, Comput. Methods Appl. Mech. Engrg., 300, 716-733, 2016 · Zbl 1425.65168 · doi:10.1016/j.cma.2015.12.005 |
| [43] | Lehrenfeld, C.; Olshanskii, M., An Eulerian finite element method for PDEs in time-dependent domains, ESAIM Math. Model. Numer. Anal., 53, 2, 585-614, 2019 · Zbl 1422.65223 · doi:10.1051/m2an/2018068 |
| [44] | Li, R., On multi-mesh \(H\)-adaptive methods, J. Sci. Comput., 24, 3, 321-341, 2005 · Zbl 1080.65111 · doi:10.1007/s10915-004-4793-5 |
| [45] | Li, R., Liu, Q., Yang, F. : A reconstructed discontinuous approximation on unfitted meshes to \(H({\rm curl})\) and \(H({\rm div})\) interface problems, Comput. Methods Appl. Mech. Engrg. 403, no. part A, Paper No. 115723, 27 (2023) · Zbl 1536.65148 |
| [46] | Li, R.; Yang, F., A discontinuous Galerkin method by patch reconstruction for elliptic interface problem on unfitted mesh, SIAM J. Sci. Comput., 42, 2, A1428-A1457, 2020 · Zbl 1440.65218 · doi:10.1137/19M1290528 |
| [47] | Li, Z., The immersed interface method using a finite element formulation, Appl. Numer. Math., 27, 3, 253-267, 1998 · Zbl 0936.65091 · doi:10.1016/S0168-9274(98)00015-4 |
| [48] | H. Liu, L. Zhang, X. Zhang, and W. Zheng, Interface-penalty finite element methods for interface problems in \(H^1\), H(curl), and H(div), Comput. Methods Appl. Mech. Engrg. 367, 113137, 16 (2020) · Zbl 1442.65382 |
| [49] | Lu, P.; Chen, H.; Qiu, W., An absolutely stable \(hp\)-HDG method for the time-harmonic Maxwell equations with high wave number, Math. Comp., 86, 306, 1553-1577, 2017 · Zbl 1362.78012 · doi:10.1090/mcom/3150 |
| [50] | Lu, P.; Wu, H.; Xu, X., Continuous interior penalty finite element methods for the time-harmonic Maxwell equation with high wave number, Adv. Comput. Math., 45, 5-6, 3265-3291, 2019 · Zbl 1435.78020 · doi:10.1007/s10444-019-09737-2 |
| [51] | Massing, A.; Larson, MG; Logg, A.; Rognes, ME, A stabilized Nitsche fictitious domain method for the Stokes problem, J. Sci. Comput., 61, 3, 604-628, 2014 · Zbl 1417.76028 · doi:10.1007/s10915-014-9838-9 |
| [52] | Melenk, Jens M.; Sauter, Stefan A., Wavenumber-explicit \(hp\)-FEM analysis for Maxwell’s equations with transparent boundary conditions, Found. Comput. Math., 21, 1, 125-241, 2021 · Zbl 1460.35085 · doi:10.1007/s10208-020-09452-1 |
| [53] | Monk, P., Finite element methods for Maxwell’s equations, 2003, New York: Numerical Mathematics and Scientific Computation. Oxford University Press, New York · Zbl 1024.78009 · doi:10.1093/acprof:oso/9780198508885.001.0001 |
| [54] | Nédélec, J-C, Mixed finite elements in \({ R}^3\), Numer. Math., 35, 3, 315-341, 1980 · Zbl 0419.65069 · doi:10.1007/BF01396415 |
| [55] | Nédélec, J-C, A new family of mixed finite elements in \({ R}^3\), Numer. Math., 50, 1, 57-81, 1986 · Zbl 0625.65107 · doi:10.1007/BF01389668 |
| [56] | Nguyen, NC; Peraire, J.; Cockburn, B., Hybridizable discontinuous Galerkin methods for the time-harmonic Maxwell’s equations, J. Comput. Phys., 230, 19, 7151-7175, 2011 · Zbl 1230.78031 · doi:10.1016/j.jcp.2011.05.018 |
| [57] | Perugia, I.; Schötzau, D., The \(hp\)-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations, Math. Comp., 72, 243, 1179-1214, 2003 · Zbl 1084.78007 · doi:10.1090/S0025-5718-02-01471-0 |
| [58] | Perugia, I.; Schötzau, D.; Monk, P., Stabilized interior penalty methods for the time-harmonic Maxwell equations, Comput. Methods Appl. Mech. Engrg., 191, 41-42, 4675-4697, 2002 · Zbl 1040.78011 · doi:10.1016/S0045-7825(02)00399-7 |
| [59] | Preuss, J. : Higher order unfitted isoparametric space-time FEM on moving domains, Master’s thesis. University of Göttingen (2018) |
| [60] | Sármány, D.; Izsák, F.; van der Vegt, JJW, Optimal penalty parameters for symmetric discontinuous Galerkin discretisations of the time-harmonic Maxwell equations, J. Sci. Comput., 44, 3, 219-254, 2010 · Zbl 1203.78046 · doi:10.1007/s10915-010-9366-1 |
| [61] | Saye, RI, High-order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles, SIAM J. Sci. Comput., 37, 2, A993-A1019, 2015 · Zbl 1328.65070 · doi:10.1137/140966290 |
| [62] | Schatz, AH, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp., 28, 959-962, 1974 · Zbl 0321.65059 · doi:10.1090/S0025-5718-1974-0373326-0 |
| [63] | Strouboulis, T.; Babuška, I.; Copps, K., The design and analysis of the generalized finite element method, Comput. Methods Appl. Mech. Engrg., 181, 1-3, 43-69, 2000 · Zbl 0983.65127 · doi:10.1016/S0045-7825(99)00072-9 |
| [64] | von Wahl, H.; Richter, T.; Lehrenfeld, C., An unfitted Eulerian finite element method for the time-dependent Stokes problem on moving domains, IMA J. Numer. Anal., 42, 3, 2505-2544, 2022 · Zbl 1502.65147 · doi:10.1093/imanum/drab044 |
| [65] | Wei, Z.; Li, C.; Zhao, S., A spatially second order alternating direction implicit (ADI) method for solving three dimensional parabolic interface problems, Comput. Math. Appl., 75, 6, 2173-2192, 2018 · Zbl 1409.65059 · doi:10.1016/j.camwa.2017.06.037 |
| [66] | Wu, H.; Xiao, Y., An unfitted \(hp\)-interface penalty finite element method for elliptic interface problems, J. Comput. Math., 37, 3, 316-339, 2019 · Zbl 1449.65326 · doi:10.4208/jcm.1802-m2017-0219 |
| [67] | Yang, F., Xie, X. : An unfitted finite element method by direct extension for elliptic problems on domains with curved boundaries and interfaces. J. Sci. Comput. 93, no. 3, Paper No. 75, 26 (2022) · Zbl 1503.65303 |
| [68] | Zhong, L.; Shu, S.; Wittum, G.; Xu, J., Optimal error estimates for Nedelec edge elements for time-harmonic Maxwell’s equations, J. Comput. Math., 27, 5, 563-572, 2009 · Zbl 1212.65467 · doi:10.4208/jcm.2009.27.5.011 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
