Two-grid methods for Maxwell’s equations in a Cole-Cole dispersive medium. (English) Zbl 07878413
Summary: A two-grid method (TGM) for the time-dependent Maxwell’s equations in Cole-Cole dispersive media with a fractional time derivative term is proposed. We employ the lowest Raviart-Thomas-Nédélec mixed finite elements to discrete the space. It is known that for these type of Nédélec edge finite elements, the standard TGM cannot be applied directly. Therefore, we modified the traditional TGM, and the discrete process can be divided into two steps. Firstly, we get the rough discrete solutions on the coarse mesh. At the same time, superconvergence results can be obtained by using a post-processing technique. Secondly, the superconvergent solutions on the coarse grid are added on the fine mesh as a correction, and the optimal error estimates could be obtained accordingly. Finally, the numerical experiments can verify that the theoretical results are correct and reasonable.
MSC:
| 65-XX | Numerical analysis |
| 35Q61 | Maxwell equations |
| 35R11 | Fractional partial differential equations |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |
Keywords:
Maxwell’s equations; two-grid method; Raviart-Thomas-Nédélec mixed finite elements; postprocessing techniqueReferences:
| [1] | X.X. Bai and H.X. Rui, An efficient FDTD algorithm for 2d/3d time fractional Maxwell’s system, Appl. Math. Lett. 116, 106992 (2021). · Zbl 1473.78014 |
| [2] | X.X. Bai, S. Wang and H.X. Rui, Numerical analysis of finite-difference time-domain method for 2d/3d Maxwell’s equations in a Cole-Cole dispersive medium, Comput. Math. Appl. 93, 230-252 (2021). · Zbl 1524.65312 |
| [3] | G. Bao, M.M. Zhang, X. Jiang, P.J. Li and X.K. Yuan, An adaptive finite element DtN method for Maxwell’s equations, East Asian J. Appl. Math. 13, 610-645 (2023). · Zbl 1527.78012 |
| [4] | H.Z. Cai, X.Y. Hu, B. Xiong and M.S. Zhdanov, Finite-element time-domain modeling of electro-magnetic data in general dispersive medium using adaptive Padé series, Comput. Geosci. 109, 194-205 (2017). |
| [5] | Y.P. Chen, Superconvergence of mixed finite element methods for optimal control problems, Math. Comput. 77, 1269-1291 (2008). · Zbl 1193.49029 |
| [6] | Y.P. Chen and H.Z. Hu, Two-grid method for miscible displacement problem by mixed finite ele-ment methods and mixed finite element method of characteristics, Commun. Comput. Phys. 19, 1503-1528 (2016). · Zbl 1373.76079 |
| [7] | B. Cockburn, G. Kanschat, I. Perugia and D. Schötzau, Superconvergence of the local discontinu-ous Galerkin method for elliptic problems on Cartesian grids, SIAM J. Numer. Anal. 39, 264-285 (2001). · Zbl 1041.65080 |
| [8] | K.S. Cole and R.H. Cole, Dispersion and absorption in dielectrics I. Alternating current charac-teristics, J. Chem. Phys. 9, 341-351 (1941). |
| [9] | C.N. Dawson, M.F. Wheeler and C.S. Woodward, A two-grid finite difference scheme for nonlin-ear parabolic equations, SIAM J. Numer. Anal. 35, 435-452 (1998). · Zbl 0927.65107 |
| [10] | S. Gabriel, R.W. Lau and C. Gabriel, The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues, Phys. Med. Biol. 41, 2271-2293 (1996). |
| [11] | Y.J. Gong, C.J. Chen, Y.Z. Lou and G.Y. Xue, Crank-Nicolson method of a two-grid finite volume element algorithm for nonlinear parabolic equations, East Asian J. Appl. Math. 11, 540-559 (2021). · Zbl 1475.65168 |
| [12] | E.R. Harrell and N. Nakajima, Modified Cole-Cole plot based on viscoelastic properties for char-acterizing molecular architecture of elastomers, J. Appl. Poly. Sci. 29, 995-1010 (1984). |
| [13] | C.B. Huang and M. Stynes, Superconvergence of a finite element method for the multi-term time-fractional diffusion problem, J. Sci. Comput. 82, 1-17 (2020). · Zbl 1447.65078 |
| [14] | C. Huang and L.L. Wang, An accurate spectral method for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media, Adv. Comput. Math. 45, 707-734 (2019). · Zbl 1415.78011 |
| [15] | Y.Q. Huang and Y.P. Chen, A multilevel iterate correction method for solving nonlinear singular problems, Natural Journal of Xiangtan University 16, 23-26 (1994). · Zbl 0802.65095 |
| [16] | Y.Q. Huang, J.C. Li, W. Yang and S.Y. Sun, Superconvergence of mixed finite element approxima-tions to 3-D Maxwell’s equations in metamaterials, J. Comput. Phys. 230, 8275-8289 (2011). · Zbl 1230.78029 |
| [17] | T. Karacolak, E.C. Moreland and E. Topsakal, Cole-Cole model for glucose-dependent dielectric properties of blood plasma for continuous glucose monitoring, Microw. Opt. Technol. Lett. 55, 1160-1164 (2013). |
| [18] | J.C. Li, Unified analysis of leap-frog methods for solving time-domain Maxwell’s equations in dispersive media, J. Sci. Comput. 47, 1-26 (2011). · Zbl 1230.78030 |
| [19] | J.C. Li, Y.Q. Huang and Y.P. Lin, Developing finite element methods for Maxwell’s equations in a Cole-Cole dispersive medium, SIAM J. Sci. Comput. 33, 3153-3174 (2011). · Zbl 1238.65097 |
| [20] | J.F. Lin and Q. Lin, Global superconvergence of the mixed finite element methods for 2-D Maxwell equations, J. Comput. Math. 21, 637-646 (2003). · Zbl 1032.65101 |
| [21] | Q. Lin and J.C. Li, Superconvergence analysis for Maxwell’s equations in dispersive media, Math. Comput. 77, 757-771 (2008). · Zbl 1136.78015 |
| [22] | Q. Lin and J.F. Lin, Finite Element Methods: Accuracy and Improvement, Beijing Science Press (2006). |
| [23] | Q. Lin and N.N. Yan, Global superconvergence for Maxwell’s equations, Math. Comput. 69, 159-176 (2000). · Zbl 0945.65118 |
| [24] | Y.M. Lin and C.J. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. 225, 1533-1552 (2007). · Zbl 1126.65121 |
| [25] | M. Mandal and G. Nelakanti, Superconvergence results for linear second-kind Volterra integral equations, J. Appl. Math. Comput. 57, 247-260 (2018). · Zbl 1394.65167 |
| [26] | P. Monk, An analysis of Nédélec’s method for the spatial discretization of Maxwell’s equations, J. Comp. Appl. Math. 47, 101-121 (1993). · Zbl 0784.65091 |
| [27] | P. Monk, Finite Element Methods for Maxwell’s Equations, Oxford University Press (2003). · Zbl 1024.78009 |
| [28] | P. Monk, Superconvergence of finite element approximations to Maxwell’s equations, Numer. Meth. Part. D. E. 10, 793-812 (2010). · Zbl 0809.65124 |
| [29] | M. Mu and J.C. Xu, A two-grid method of a mixed Stokes-Darcy model for coupling fluid flow with porous media flow, SIAM J. Numer. 45, 1801-1813 (2007). · Zbl 1146.76031 |
| [30] | J.C. Nédélec, Mixed finite elements in 3 , Numer. Math. 35, 315-341 (1980). · Zbl 0419.65069 |
| [31] | K. Oldham and J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Elsevier (1974). · Zbl 0292.26011 |
| [32] | I.T. Rekanos and T.G. Papadopoulos, An auxiliary differential equation method for FDTD model-ing of wave propagation in Cole-Cole dispersive media, IEEE T. Antenn. Propag. 58, 3666-3674 (2010). · Zbl 1369.78819 |
| [33] | A. Revil, P. Leroy, A. Ghorbani, N. Florsch and A.R. Niemeijer, Compaction of quartz sands by pressure solution using a Cole-Cole distribution of relaxation times, J. Geophy. Res. 111, B09205 (2006). |
| [34] | J.W. Schuster and R.J. Luebbers, An FDTD algorithm for transient propagation in biological tissue with a Cole-Cole dispersion relation, IEEE Antenn. Propag. Soc. Int. Symp. 4, 1988-1991 (1998). |
| [35] | D.Y. Shi and H.J. Yang, Unconditional optimal error estimates of a two-grid method for semilinear parabolic equation, Appl. Math. Comput. 310, 40-47 (2017). · Zbl 1427.65184 |
| [36] | Z.Z. Sun and X.N. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math. 56, 193-209 (2006). · Zbl 1094.65083 |
| [37] | Z.K. Tian, Y.P. Chen and J.Y. Wang, Superconvergence analysis of bilinear finite element for the nonlinear Schrödinger equation on the rectangular mesh, Adv. Appl. Math. Mech. 10, 468-484 (2018). · Zbl 1488.65459 |
| [38] | F. Torres, P. Vaudon and B. Jecko, Application of fractional derivatives to the FDTD modeling of pulse propagation in a Cole-Cole dispersive medium, Microw. Opt. Techn. Lett. 13, 300-304 (1996). |
| [39] | G.M. Tsangaris, G.C. Psarras and N. Kouloumbi, Electric modulus and interfacial polarization in composite polymeric systems, J. Mater. Sci. 33, 2027-2037 (1998). |
| [40] | J.X. Wang, J.W. Zhang and Z.M. Zhang, A CG-DG method for Maxwell’s equations in Cole-Cole dispersive media, J. Comput. Appl. Math. 393, 113480 (2021). · Zbl 1468.65153 |
| [41] | L.X. Wang, Q. Zhang and Z.M. Zhang, Superconvergence analysis and PPR recovery of arbitrary order edge elements for Maxwell’s equations, J. Sci. Comput. 78, 1207-1230 (2019). · Zbl 1417.65208 |
| [42] | Y. Wang, Y.P. Chen, Y.Q. Huang and Y. Liu, Two-grid methods for semi-linear elliptic interface problems by immersed finite element methods, Appl. Math. Mech. 40, 1657-1676 (2019). · Zbl 1431.65220 |
| [43] | C. Wu, Y.Q. Huang, N.Y. Yi and J.Y. Yuan, Superconvergent recovery of edge finite element ap-proximation for Maxwell’s equations, J. Comput. Appl. Math. 371, 113302 (2020). · Zbl 1506.65231 |
| [44] | J.H. Xiao, Z.M. Zhu and X.P. Xie, Semi-discrete and fully discrete weak Galerkin finite element methods for a quasistatic Maxwell viscoelastic model, Numer. Math. Theor. Meth. Appl. 16, 79-110 (2023). · Zbl 1524.35640 |
| [45] | J.C. Xu, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput. 15, 231-237 (1994). · Zbl 0795.65077 |
| [46] | J.C. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal. 33, 1759-1777 (1996). · Zbl 0860.65119 |
| [47] | J.C. Xu and A.H. Zhou, A two-grid discretization scheme for eigenvalue problems, Math. Comput. 70, 17-25 (2001). · Zbl 0959.65119 |
| [48] | C.H. Yao, F.R. Li and Y.M. Zhao, Superconvergence analysis of two-grid FEM for Maxwell’s equa-tions with a thermal effect, Comput. Math. Appl. 79, 3378-3393 (2020). · Zbl 1446.65183 |
| [49] | Y.M. Zhao, Y.D. Zhang, F.W. Liu, I. Turner, Y.F. Tang and V. Anh, Convergence and supercon-vergence of a fully-discrete scheme for multi-term time fractional diffusion equations, Comput. Math. Appl. 73, 1087-1099 (2017). · Zbl 1412.65159 |
| [50] | J. Zhou, X. Hu, L. Zhong, S. Shu and L. Chen, Two-grid methods for Maxwell eigenvalue prob-lems, SIAM. J. Numer. Anal. 52, 2027-2047 (2014). · Zbl 1304.78015 |
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