ANALYSIS OF A QUADRILATERAL EDGE ELEMENT METHOD FOR MAXWELL EQUATIONS

doi:10.1007/s10473-026-0116-3

Abstract

Abstract: A new quadrilateral edge element method is proposed and analyzed for Maxwell equations. This proposed method is based on Duan-Liang quadrilateral element (Math. Comp. 73(2004), pp. 1-18). When applied to the eigenvalue problem, the method is spectral-correct and spurious-free. Stability and error estimates are obtained, including the interpolation error estimates and the error estimates between the finite element solution and the exact solution. The method is suitable for singular solution as well as smooth solution, and consequently, the method is valid for nonconvex domains which may have a number of reentrant corners. Of course, the method is suitable for arbitrary quadrilaterals (under the usual shape-regular condition).

Key words: Maxwell equations, finite element method, quadrilateral mesh, stability, error bound, spectral approximation

CLC Number:

65N30

Zhijie DU¹, Huoyuan DUAN^2,*, Caihong WANG². ANALYSIS OF A QUADRILATERAL EDGE ELEMENT METHOD FOR MAXWELL EQUATIONS[J].Acta mathematica scientia,Series B, 2026, 46(1): 275-292.

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References

[1] Adams R A. Sobolev Spaces.New York: Academic Press, 1975
[2] Amrouche C, Bernardi C, Dauge M, et al. Vector potentials in three-dimensional non-smooth domains. Math Methods Appl Sci, 1998, ${\bf 21}$ 823-864
[3] Arnold D N, Boffi D, Falk R S. Quadrilateral H(div) finite elements. SIAM J Numer Anal, 2005, ${\bf 42}$ 2429-2451
[4] Babuška I, Osborn J E. Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems. Math Comp, 1989, ${\bf 52}$ 275-297
[5] Babuska I, Osborn J.Eigenvalue problems//Ciarlet P G, Lions J L, eds. Handbook of Numerical Analysis II. Amsterdam: North-Holland, 1991: 641-787
[6] Bernardi C, Girault V.A local regularization operator for triangular and quadrilateral finite elements. SIAM J Numer Anal, 1998, ${\bf 35}$ 1893-1916
[7] Birman M, Solomyak M. $L^2$-theory of the Maxwell operator in arbitrary domains. Russ Math Surv, 1987, ${\bf 42}$ 75-96
[8] Bochev P B, Ridzal D. Rehabilitation of the lowest-order Raviart-Thomas element on quadrilateral grids. SIAM J Numer Anal, 2008, ${\bf 47}$ 487-507
[9] Boffi D. Approximation of eigenvalues in mixed form, discrete compactness property,application to hp mixed finite elements. Comput Methods Appl Mech Engrg, 2007, ${\bf 196}$ 3672-3681
[10] Boffi D. Finite element approximation of eigenvalue problems. Acta Numerica, 2010, ${\bf 19}$ 1-120
[11] Boffi D, Fernandes P, Gastaldi L, et al. Computational models of electromagnetic resonators: Analysis of edge element approximation. SIAM J Numer Anal, 1999, ${\bf 36}$ 1264-1290
[12] Boffi D, Brezzi F, Gastaldi L. On the convergence of eigenvalues for mixed formulations. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 1997, ${\bf 25(1)}$ 131-154
[13] Boffi D, Kikuchi F, Schöberl J. Edge element computation of Maxwell's eigen-values on general quadrilateral meshes. Math Models Methods Appl Sci, 2006, ${\bf 16}$ 265-273
[14] Bossavit A.Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements. New York: Academic Press, 1998
[15] Brenner S C, Scott L R.The Mathematical Theory of Finite Element Methods. Berlin: Springer-Verlag, 1996
[16] Brenner S C, Li F, Sung L Y. Nonconforming Maxwell eigensolvers. J Sci Comput, 2009, ${\bf 40}$ 51-85
[17] Brezzi F, Fortin M.Mixed and Hybrid Finite Element Methods. New York: Springer-Verlag, 1991
[18] Buffa A, Perugia I. Discontinuous Galerkin approximation of the Maxwell eigenproblem. SIAM J Numer Anal, 2006, ${\bf 44}$ 2198-2226
[19] Buffa A, Ciarlet Jr P, Jamelot E. Solving electromagnetic eigenvalue problems in polyhedral domains. Numer Math, 2009, ${\bf 113}$ 497-518
[20] Caorsi S, Fernandes P, Raffetto M. On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems. SIAM J Numer Anal, 2000, ${\bf 38}$ 580-607
[21] Caorsi S, Fernandes P, Raffetto M. Spurious-free approximations of electromagnetic eigenproblems by means of Nédléc-type elements. ESAIM: Math Modell Numer Anal, 2001, ${\bf 35}$ 331-354
[22] Cessenat M.Mathematical Methods in Electromagnetism: Linear Theory and Applications. Singapore: World Scientific, 1996
[23] Chatelin F. Convergence of approximation methods to compute eigenelements of linear operations. SIAM J Numer Anal, 1973, ${\bf 10}$ 939-948
[24] Chatelin F.Spectral Approximations of Linear Operators. New York: Academic Press 1983
[25] Ciarlet P G.Basic error estimates for elliptic problems//Ciarlet P G, Lions J L, eds. Handbook of Numerical Analysis, Vol. II, Finite Element Methods (part 1). Amsterdam: North-Holland, 1991: 17-351
[26] Costabel M. A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains. Math Methods Appl Sci, 1990, ${\bf 12}$ 365-368
[27] Costabel M. A coercive bilinear form for Maxwell's equations. J Math Anal Appl, 1991, ${\bf 157}$ 527-541
[28] Costabel M, Dauge M . Singularities of electromagnetic fields in polyhedral domains. Arch Rational Mech Anal, 2000, ${\bf 151}$ 221-276
[29] Costabel M, Dauge M. Maxwell and Lamé eigenvalues on polyhedra. Math Meth Appl Sci, 1999, ${\bf 22}$ 243-258
[30] Descloux J, Nassif N, Rappaz J. On spectral approximation, Part 1. The problem of convergence. RAIRO Anal Numér.1978, ${\bf 12}$ 97-112
[31] Descloux J, Nassif N, Rappaz J. On spectral approximation Part 2. Error estimates for the Galerkin method convergence. RAIRO Anal Numér, 1978, ${\bf 12}$ 113-119
[32] Duan H Y, Du Z J, Liu W, et al. New mixed elements for Maxwell equations. SIAM J Numer Anal, 2019, ${\bf 57}$ 320-354
[33] Duan H Y, Jia F, Lin P, et al. The local $L^2$ projected $C^0$ finite element method for Maxwell problem. SIAM J Numer Anal, 2009, ${\bf 47}$ 1274-1303
[34] Duan H Y, Li S, Tan R C E, et al. A delta-regularization finite element method for a double curl problem with divergence-free constraint. SIAM J Numer Anal, 2012, ${\bf 50}$ 3208-3230
[35] Duan H Y, Liang G P. Nonconforming elements in least-squares mixed finite element methods. Math Comp, 2004, ${\bf 73}$ 1-18
[36] Duan H Y, Liang G P. A locking-free Reissner-Mindlin quadrilateral element. Math Comp, 2004, ${\bf 73}$ 1655-1671
[37] Duan H Y, Lin P, Saikrishnan P, et al. A least squares finite element method for the magnetostatic problem in a multiply-connected Lipschitz domain. SIAM J Numer Anal, 2007, ${\bf 45}$ 2537-2563
[38] Duan H Y, Lin P, Tan R C E. Error estimates for a vectorial second-order elliptic eigenproblem by the local $L^2$ projected $C^0$ finite element method. SIAM Journal on Numerical Analysis, 2013, ${\bf 51}$ 1678-1714
[39] Duan H Y, Lin P, Tan R C E. A finite element method for a curlcurl-graddiv eigenvalue interface problem. SIAM J Numer Anal, 2016, ${\bf 54}$ 1193-1228
[40] Fernandes P, Gilardi G. Magnetostatic and Electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Math Models and Methods Appl Sci, 1997, ${\bf 7}$ 957-991
[41] Girault V. A local projection operator for quadrilateral finite elements. Math Comp, 1995, ${\bf 64}$ 1421-1431
[42] Girault V, Raviart P A.Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Berlin: Springer-Verlag, 1986
[43] Hiptmair R.Finite elements in computational electromagnetism. Acta Numerica, 2002: 237-339
[44] Houston P, Perugia I, Schneebeli A, et al. Interior penalty method for indefinitetime-harmonic Maxwell equations. Numer Math, 2005, ${\bf 100}$ 485-518
[45] Jin J M.The Finite Element Method in Electromagnetics (2nd Edition). New York: John Wiley & Sons, 2002
[46] Kato T.Perturbation Theory of Linear Operators. Berlin: Springer-Verlag, 1966
[47] Kikuchi F. Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism. Comput Methods Appl Mech Engrg, 1987, ${\bf 64}$ 509-521
[48] Mercier B, Osborn J, Rappaz J, et al. Eigenvalue approximation by mixed and hybrid methods. Math Comput, 1981, ${\bf 36}$ 427-453
[49] Monk P.Finite Element Methods for Maxwell Equations. Oxford: Clarendon Press, 2003
[50] Nédélec J C. Mixed finite elements in $R^3$. Numer Math, 1980, ${\bf 35}$ 315-341
[51] Nédélec J C. A new family of mixed finite elements in $R^3$. Numer Math, 1986, ${\bf 50}$ 57-81
[52] Nicaise S. Edge elements on anisotropic meshes and approximation of the Maxwell equations. SAIM J Numer Anal, 2001, ${\bf 39}$ 784-816
[53] Raviart R A, Thomas J M.A mixed finite element method for 2nd order elliptic problems//Galligani I, Magenes E, eds. Mathematical Aspects of Finite Element Methods. New York: Springer-Verlag, 1977: 292-315
[54] Roberts J E, Thomas J M.Mixed and hybrid methods//Ciarlet P G, Lions J L, eds. Handbook of Numerical Analysis. Amsterdam: North-Holland, 1991: 523-639
[55] Scott L R, Zhang S. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math Comp, 1990, ${\bf 54}$ 483-493
[56] Weber C. A local compactness theorem for Maxwell's equations. Math Meth Appl Sci, 1990, ${\bf 2}$ 12-25
[57] Wang J, Mathew T.Mixed finite element methods over quadrilaterals//Dimov J T, Sendov B, Vassilevski P, eds. Proceedings of 3rd Internat Conference on Advances in Numerical Methods and Applications. River Edge, NJ: World Scientific, 1994: 203-214