Hodge decomposition for two-dimensional time-harmonic Maxwell’s equations: impedance boundary condition. (English) Zbl 1361.78007
This paper deals with the study of 2D time-harmonic Maxwell equations in relation to the transverse magnetic problem. In the case of problems on nonconvex domains or problems with discontinuity in the material properties, optimal convergence can be restored by using adaptive algorithms based on the Hodge decomposition method. The authors also discuss the well-posedness of the cavity problem when both the magnetic permeability and the electric permittivity can change the sign. Some numerical experiments illustrate the case of the Maxwell equations with inhomogeneous and anisotropic pemittivity, sign-changing permeability, and the impedance boundary condition.
Reviewer: Teodora-Liliana Rădulescu (Craiova)
MSC:
| 78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |
| 58A14 | Hodge theory in global analysis |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 35Q61 | Maxwell equations |
| 78A25 | Electromagnetic theory (general) |
Keywords:
finite element method; Hodge decomposition; time-harmonic Maxwell equation; metamaterial; electromagnetic cloakingReferences:
| [1] | SeniorT, VolakisJ. Approximate Boundary Condition in Electromagnetics. IEEE Press: New York, 1995. · Zbl 0828.73001 |
| [2] | MonkP. Finite Element Methods for Maxwell’s Equations. Oxford University Press: New York, 2003. · Zbl 1024.78009 |
| [3] | BrennerS, CuiJ, NanZ, SungLY. Hodge decomposition for divergence‐free vector fields and two‐dimensional Maxwell’s equations. Mathematics of Computation2012; 81:643-659. · Zbl 1274.78089 |
| [4] | BrennerS, GedickeJ, SungLY. An adaptive P_1 finite element method for two‐dimensional Maxwell’s equations. Journal of Scientific Computing2013; 55:738-754. · Zbl 1266.78027 |
| [5] | CuiJ. Multigrid methods for two‐dimensional Maxwell’s equations on graded meshes. Journal of Computational and Applied Mathematics2014; 255:231-247. · Zbl 1291.78055 |
| [6] | NaderE, ZiolkowskiR. Metamaterials: Physics and Engineering Explorations. Wiley & Sons: Hoboken, N.J., 2006. |
| [7] | SolymarL, ShamoninaE.. Waves in Metamaterials. Oxford University Press: Oxford, 2009. |
| [8] | FernandesP, RaffettoM. Existence, uniqueness and finite element approximation of the solution of time‐harmonic electromagnetic boundary value problems involving metamaterials. The International Journal for Computation and Mathematics in Electrical and Electronic Engineering2005; 24:1450-1469. · Zbl 1079.78033 |
| [9] | FernandesP, RaffettoM. Well‐posedness and finite element approximability of time‐harmonic electromagnetic boundary value problems involving bianisotropic materials and metamaterials. Mathematical Models & Methods in Applied Sciences2009; 19:2299-2335. · Zbl 1205.78058 |
| [10] | CostabelM, DarrigrandE, SaklyH. The essential spectrum of the volume integral operator in electromagnetic scattering by a homogeneous body. Comptes Rendus Mathématique. Académie des Sciences, Paris2012; 350:193-197. · Zbl 1247.78011 |
| [11] | ChesnelL, CiarletPJr. Compact imbeddings in electromagnetism with interfaces between classical materials and metamaterials. SIAM Journal on Mathematical Analysis2011; 43:2150-2169. · Zbl 1250.78007 |
| [12] | Bonnet‐Ben DhiaAS, ChesnelL. Ciarlet P Jr. T‐coercivity for scalar interface problems between dielectrics and metamaterials. ESAIM Mathematical Modelling and Numerical Analysis2012; 46:1363-1387. · Zbl 1276.78008 |
| [13] | Bonnet‐Ben DhiaAS, ChesnelL, CiarletPJr. Two‐dimensional Maxwell’s equations with sign‐changing coefficients. Applied Numerical Mathematics2014; 79:29-41. · Zbl 1291.78018 |
| [14] | Bonnet‐Ben DhiaAS, ChesnelL. T‐coercivity for the Maxwell problem with sign‐changing coefficients. Communications in Partial Differential Equations2014; 39:1007-1031. · Zbl 1297.35229 |
| [15] | CiarletP. The Finite Element Method for Elliptic Problems. North‐Holland: Amsterdam, 1978. · Zbl 0383.65058 |
| [16] | BrennerS, ScottLR. The Mathematical Theory of Finite Element Methods, third edition. Springer‐Verlag: New York, 2008. · Zbl 1135.65042 |
| [17] | GiraultV, RaviartPA. Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer‐Verlag: Berlin, 1986. · Zbl 0585.65077 |
| [18] | ReedM, SimonB. Methods of Modern Mathematical Physics. I. Functional Analysis. Academic Press: New York, 1972. · Zbl 0242.46001 |
| [19] | Nec̆asJ. Direct Methods in the Theory of Elliptic Equations. Springer: Heidelberg, 2012. · Zbl 1246.35005 |
| [20] | LaxP. Functional Analysis. Wiley‐Interscience: New York, 2002. |
| [21] | CostabelM, DaugeM, NicaiseS. Singularities of Maxwell interface problems. ESAIM Mathematical Modelling and Numerical Analysis1999; 33:627-649. · Zbl 0937.78003 |
| [22] | HörmanderL. The Analysis of Linear Partial Differential Operators. III. Springer‐Verlag: Berlin, 1985. · Zbl 0601.35001 |
| [23] | HazardC, LenoirM. On the solution of time‐harmonic scattering problems for Maxwell’s equations. SIAM Journal on Mathematical Analysis1996; 27:1597-1630. · Zbl 0860.35129 |
| [24] | DaugeM. Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Mathematics 1341. Springer‐Verlag: Berlin‐Heidelberg, 1988. · Zbl 0668.35001 |
| [25] | GrisvardP. Elliptic Problems in Nonsmooth Domains. Pitman: Boston, 1985. · Zbl 0695.35060 |
| [26] | NazarovS, PlamenevskyB. Elliptic Problems in Domains with Piecewise Smooth Boundaries. de Gruyter: Berlin‐New York, 1994. · Zbl 0806.35001 |
| [27] | SchatzA. An observation concerning Ritz-Galerkin methods with indefinite bilinear forms. Mathematics of Computation1974; 28:959-962. · Zbl 0321.65059 |
| [28] | LiJ, HuangY, YangW. An adaptive edge finite element method for electromagnetic cloaking simulation. Journal of Computational Physics2013; 249:216-232. · Zbl 1304.78014 |
| [29] | Nédélec JC. Mixed finite elements in R^3. Numerische Mathematik1980; 35:315-341. · Zbl 0419.65069 |
| [30] | PendryJ, SchurigD, SmithD. Controlling electromagnetic fields. Science2006; 312:1780-1782. · Zbl 1226.78003 |
| [31] | NicaiseS, VenelJ. A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients. Journal of Computational and Applied Mathematics2011; 235:4272-4282. · Zbl 1220.65167 |
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.
