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Recent Advances in Mathematical Analysis andNumerical Simulation of Invisibility Cloaks with

Metamaterials

Yunqing HuangXiangtan University, China

Collaborators: Jichun Li (Univ of Nevada Las Vegas),Wei Yang (Xiangtan U)

Yunqing Huang (Xiangtan U.) July 18, 2019 1 / 69

1 Introduction to metamaterials and cloaking with metamaterials

2 Carpet cloak model

3 Other cloaking models and applications

4 Superconvergence of edge elements

5 Major References






5 Major References


Metamaterials: historical milestone

Figure 1: (Left)Veselago, (Middle) Pendry, (Right) Smith

• In 1968, Veselago studied the EM material with both ε < 0 and µ < 0.(Sov. Phys. USPEKHI, 1968, 10: 509-514.)• Pendry introduced the idea of designing electric (ε < 0, Phys. Rev.Lett., 1996,76: 4773) and magnetic (µ < 0,Microwave Theory Tech.,1999, 47: 2075) plasmon .• Until 2000, D.R. Smith et al (UCSD) constructed DNG(Science,2001, 292: 77) inspired by J. Pendry et al’s work.


Metamaterials

What is the metamaterials?

Figure 2: Metamaterials


Metamaterials

Metamaterials: an arrangement of artificial structural elements,designed to achieve advantageous and unusual electromagneticproperties.Effective medium description using Maxwell equations with thepermittivity ε, permeability µ , refraction n.

Figure 3: Periodic structures


Metamaterials

Metamaterials• have lattice constants smaller than the wavelength (λ

a > 1).• are artificial materials.• can be treated as homogeneous media(ε,µ).• can have a negative index of refraction (n < 0).

Exotic electromagnetic properties: negative refraction index;negative phase velocity; reversed Doppler effects; re-focusing,...


Negative index materials

How to realize a negative permittivity (ε < 0)?For a lossless metal rod:

εeff = 1−ω2

p

ω2 ,

where ω2p =

jeff e2

ε0me, jeff =

πr2ja2 .

j : electron density, e: electron charge, me: effective mass of electrons.Using array of rods confining the electrons in space,

Figure 4: Periodic arrays of the Rods(J. P. Pendry, Phys. Rev. Lett. 1996, 76:4773)


Negative index materials

How to realize a negative permeability (µ < 0)?

Figure 5: Split ring resonator(SRR, J. P. Pendry, Microwave Theory Tech.,1999, 47: 2075)

µeff = 1−ω2

mp−ω20

ω2−ω20 + iγ

, where ω20 =

2dc20

π2r3 ,ω2mp = ω

20 (1− πr2

a2 )−1,

when ω0 < ω < ωmp, the effective permeability is negative values.Yunqing Huang (Xiangtan U.) July 18, 2019 9 / 69

Invisibility cloak with metamaterials

• Science, vol.312 (June 23, 2006): “Controlling ElectromagneticFields” (by J.B. Pendry, D. Schurig, D.R. Smith) [Cited 4510 times as2/25/15; 6115 times as 4/8/17; 6404 times as 8/26/17; 6785 times as3/15/18; 6997 times as 7/9/18; 7665 times as 7/12/19]• Science, vol.312 (June 23, 2006): “Optical Conformal Mapping” (byUlf Leonhardt). [Cited 2369 times as 2/25/15; 3110 times as 4/8/17;3250 times as 8/26/17; 3401 times as 3/15/18; 7806 times as 7/12/19]• Science, vol.314 (Nov. 10, 2006): “Metamaterial ElectromagneticCloak at Microwave Frequencies” (by Schurig, Mock, etc.) [Cited 3689times as 2/25/15; 5037 times as 4/8/17; 5289 times as 8/26/17; 5632times as 3/15/18; 5863 times as 7/9/18; 6474 times as 7/12/19]• Science, vol.328 (Apr. 16, 2010): “Three-Dimensional InvisibilityCloak at Optical Wavelengths” (by Ergin, Stenger, Brenner, Pendry,Wegener) [Cited 918 times as 3/15/18; 969 times as 7/9/18; 1063times as 7/12/19]


Mathematicians in metamaterials

• Greenleaf, Lassas and Uhlmann (2003) for conductivity equations;Approx/near cloaking: Milton, Nicorovici (May 3, 2006), Bouchitte,Schweizer (2010), Ammari etc (2013), Kohn, Weinstein etc (2008,2014), G. Bao, J. Zou, H.Y. Liu, J.Z. Li, ...

• Numer. methods: S. Nicaise (2012, 2017), Brenner, Gedicke, Sung(JSC2016, M2AS2017), J. Li, Hesthaven (JCP2014), J. Li, C.-W. Shu(CMA2017), Z. Xie etc (CiCP2016), L.-L. Wang etc (CiCP2015,CMAME2016), D. Liang etc (JSC2016), Ciarlet, Jr., P. Joly, ...


Figure 6: (A) The simulation of the cloak with the exact material properties,(B) the simulation of the cloak with the reduced material properties, (C) theexperimental measurement of the bare conducting cylinder, and (D) theexperimental measurement of the cloaked conducting cylinder. Invisible to anincident plane wave at 8.5 GHz. Source: D. Schurig et al, Science, V.314,Nov. 2006, 977-980.


Figure 7: 2D microwave cloaking structure (background image) with a plot ofthe material parameters that are implemented. Source: D. Schurig et al,Science, V.314, Nov. 2006, 977-980.Exact parameters: εz = ( b

b−a )2 r−ar ,µr = r−a

r ,µθ = rr−a

Reduced parameters: εz = ( bb−a )2,µr = ( r−a

r )2,µθ = 1.


Form invariant property for Maxwell’s equations

Theorem 1

Under a coordinate transformation x ′ = x ′(x), the Maxwell’s equations

∇×E + jωµH = 0, ∇×H− jωεE = 0, (1)

keep the same form in the transformed coordinate system:

∇′×E ′+ jωµ

′H ′ = 0, ∇′×H ′− jωε

′E ′ = 0, (2)

where all new variables are given by

E ′(x ′) = A−T E(x), H ′(x ′) = A−T H(x), A = (aij), aij =∂x ′i∂xj

, (3)

andµ′(x ′) = Aµ(x)AT/det(A), ε

′(x ′) = Aε(x)AT/det(A). (4)






5 Major References


Carpet cloak model: SIAM J Appl Math (2014)

Following Chen, Pendry, et al [Nature Communications, 2 (2011)], atriangular carpet cloak can be achieved with spatially homogeneousanisotropic dielectric materials.

Figure 8: The physical space of the carpet cloak.


Carpet cloak modeling equations

Using the mapping

x = x ′,y =H2−H1

H2y ′+

d −x ′ ·sgn(x ′)d

H1,

and the transformation optics, we can obtain the relative permittivityand permeability of the cloak given by:

ε =

[a bb c

]=

[H2

H2−H1− H1H2

(H2−H1)d sgn(x)

− H1H2(H2−H1)d sgn(x) H2−H1

H2+ H2

H2−H1(H1

d )2

],

µ =H2

H2−H1,

where sgn(x) denotes the standard sign function.


Carpet cloak modeling equations: cont’d

Diagonalizing the symmetric matrix ε as: ε = PΣPT ,

where λ1 =a+c−√

(a−c)2+4b2

2 , λ2 =a+c+

√(a−c)2+4b2

2 , and matrices Σand P are

Σ =

(λ1 00 λ2

), P =

(p1 p2p3 p4

), p2

1 + p22 = 1, p1p3 + p2p4 = 0,

and elements pi ,1≤ i ≤ 4, are given as

p1 =

√λ2−aλ2−λ1

, p2 =

√a−λ1

λ2−λ1·sgn(x),

p3 =−

√λ2−cλ2−λ1

·sgn(x), p4 =

√c−λ1

λ2−λ1.



It is easy to see that λ2 ≥ a+c+|a−c|2 ≥ a > 1, which leads to

λ1 = 1/λ2 < 1. Mapping λ1 by the lossless Drude dispersion model:

λ1(ω) = 1−ω2

p

ω2 ,

where ωp is plasma frequency, and ω is wave frequency.Substituting ε = PΣPT into D = ε0εE , we obtain

ε0E = PΣ−1PT D,

which can be written in time-domain (assuming eiωt time dependence)as follows:

ε0λ2

(E tt + ω

2p E)

= MADtt + MBD, (5)

where matrices MA and MB are

MA =

(p2

1λ2 + p22 p2p4 + p1p3λ2

p2p4 + p1p3λ2 p23λ2 + p2

4

), MB =

(p2

2 p2p4p2p4 p2

4

)ω

2p .



The governing equations for the carpet cloak:

Dt = ∇×H, (6)

ε0λ2

(E tt + ω

2p E)

= MADtt + MBD, (7)

µ0µHt =−∇×E , (8)

supplemented with initial conditions

D(x ,0) = D0(x), E(x ,0) = E0(x), H(x ,0) = H0(x), ∀ x ∈ Ω, (9)

and the perfect conducting boundary condition (PEC):

n×E = 0 on ∂ Ω. (10)

Here H denotes the magnetic field, and 2D vector and scalar curloperators:

∇×H = (∂H∂y

,−∂H∂x

)′, ∇×E =∂Ey

∂x− ∂Ex

∂y, ∀E = (Ex ,Ey )′.


Carpet cloak equations: existence

Lemma 2

The matrix MB is symmetric and non-negative definite, and the matrixMA is symmetric positive definite.

Using Laplace transforms, we can prove

Theorem 3

For any t ∈ [0,T ], there exists a unique solution(E(·, t),D(·, t),H(·, t)) ∈ (H0(curl;Ω))2×H(curl;Ω) of (6)-(10).


Stability: Li, Huang, Yang, Wood (SIAM JAM 2014)

Theorem 4

For any t ∈ [0,T ], the following stability holds true:[||∂t2E ||2 + ||∂tE ||2 + ||E ||2 + ||M

12A ∇×∂tE ||2

+||M12A ∂tD||2 + ||M

12B D||2

](t)

≤ C[||∂t2E ||2 + ||∂tE ||2 + ||E ||2 + ||M

12A ∇×∂tE ||2

+||M12A ∂tD||2 + ||M

12B D||2

](0),

where the constant C > 0 depends on the physical parametersε0,µ0,d ,H1,H2 and ωp.


Refined stability: Li, Meng, Huang (CMAM2018)

Theorem 5

For any t ∈ [0,T ], the following stability holds true:[ε0µ0µλ2(||M−

12

A ∂t2E ||2 + ω2p ||M

− 12

A ∂tE ||2) + ||∇×∂tE ||2

+||M12A ∂tD||2 + ||M

12B D||2 + ε0µ0µλ2ω

2p ||H||2

](t)

≤ C[

ε0µ0µλ2(||M−12

A ∂t2E ||2 + ω2p ||M

− 12

A ∂tE ||2) + ||∇×∂tE ||2

+||M12A ∂tD||2 + ||M

12B D||2 + ε0µ0µλ2ω

2p ||H||2

](0),

where the constant C > 0 depends on the physical parametersε0,µ0,d ,H1,H2 and ωp.


FETD scheme with edge elements

For rectangular elements K ∈ T h,

Uh = ψh ∈ L2(Ω) : ψh|K ∈Q0,0, ∀ K ∈ T h,VVV h = φh ∈ H(curl ;Ω) : φh|K ∈Q0,1×Q1,0, ∀ K ∈ T h,

where Qi ,j denotes the space of polynomials whose degrees are lessthan or equal to i and j in variables x and y , respectively.While for triangular elements, we choose

Uh = ψh ∈ L2(Ω) : ψh|K ∈ P0, ∀ K ∈ T h,VVV h = φh ∈ H(curl ;Ω) : φh|K = spanLi∇Lj −Lj∇Li, i , j = 1,2,3,∀ K ∈ T h,

where Li denotes the barycentric coordinate at vertex i of element K .The space VVV 0

h = φh ∈VVV h, n×φh = 0 on ∂ Ω is introduced to imposethe perfect conducting boundary condition n×E = 0.


The FETD scheme: cont’d

Construct a leap-frog scheme for (6)-(8): Given proper initial approximations

H0h ,D

− 12

h ,D− 3

2h ,E

− 12

h ,E− 3

2h , for n ≥ 0 find D

n+ 12

h ,En+ 1

2h ∈ V 0

h, Hn+1h ∈ Uh such that(

δτDnh,φh

)= (Hn

h ,∇×φh), (11)

ε0λ2

(δ

2τ E

n+ 12

h ,ϕh

)+ ε0λ2ω

2p

(E

n+ 12

h ,ϕh

)=

(MAδ

2τ D

n+ 12

h ,ϕh

)+

(MBD

n+ 12

h ,ϕh

), (12)

µ0µ

(δτH

n+ 12

h ,ψh

)=−(∇×E

n+ 12

h ,ψh), (13)

hold true for any φh, ϕh ∈ V 0h, ψh ∈ Uh. Difference operators:

δτun =un+ 1

2 −un− 12

τ, δ

2τ un+ 1

2 =un+ 1

2 −2un− 12 + un− 3

2

τ2 .


The FETD scheme: discrete stability

Theorem 6

For the FETD solution (Dn+ 12

h ,En+ 12

h ), denote the discrete energy attime level n:

ENGn = ε0µ0µλ2(||M−12

A δ2τ E

n+ 12

h ||2 + ω2p ||M

− 12

A δτEnh||2) + ||∇×δτE

n+ 12

h ||2

+||M12A δτDn

h||2 + ||M12B D

n+ 12

h ||2 + ε0µ0µλ2ω2p ||Hn

h ||2. (14)

Then for any n ≥ 1 and under the constraint

τ ≤min

1,

14ε0λ2

,1

4µ0µ||M−1A MB||2

,µ0µ

4,

ε0λ2

||M−1A MB||2

,h2ε0µ0µλ2

8C2inv ||M

1/2A ||2

,

(15)we have

ENGn ≤ C ·ENG0,

where C > 0 is independent of τ and h.Yunqing Huang (Xiangtan U.) July 18, 2019 26 / 69

Optimal error estimate: error equations

Define errors: un+ 1

2h = D

n+ 12

h −D(x , tn+ 12

),u = D,E , Hnh = Hn

h −H(x , tn).

The error equations are given as follows:

(δτ Dnh,φh) = (Hn

h ,∇×φh) + (∂tDn−δτDn,φh), (16)

ε0λ2(δ2τ E

n+ 12

h ,ϕh) + ε0λ2ω2p (E

n+ 12

h ,ϕh)− (MAδ2τ D

n+ 12

h ,ϕh)− (MBDn+ 1

2h ,ϕh)

= ε0λ2(∂t2En+ 12 −δ

2τ En+ 1

2 ,ϕh)− (MA(∂t2Dn+ 12 −δ

2τ Dn+ 1

2 ),ϕh), (17)

µ0µ(δτ Hn+ 1

2h ,ψh) =−(∇× E

n+ 12

h ,ψh) + µ0µ(∂tHn+ 12 −δτHn+ 1

2 ,ψh), (18)

hold true for any φh,ϕh ∈ V 0h and ψh ∈ Uh. For simplicity, we assume

that the initial conditions used for the scheme (11)–(13) are as follows:

H0h = Π2H0, D

− 12

h = ΠcD−12 , D

− 32

h = ΠcD−32 , E

− 12

h = ΠcE−12 , E

− 32

h = ΠcE−32 .


Optimal error estimate

Theorem 7Under the time step constraint (15) and the following regularityassumptions:

maxt∈[0,T ]

(||E ||Hp(curl;Ω), ||D||Hp(curl;Ω), ||∇×E ||Hp(curl;Ω)

)< ∞,

∫ T

0

(||∂t2 E ||2

Hp(curl;Ω)+ ||∂t3 E ||2

Hp(curl;Ω)+ ||∂t4 E ||2 + ||∇×∇×∂t E ||2Hp(curl;Ω)

+ ||∂t3 D||2

+||∂t4 D||2 + ||∂t D||2Hp(curl;Ω)+ ||∂t2 D||2

Hp(curl;Ω)+ ||∂t2 H||2 + ||∇×∂t3 H||2

)ds < ∞,

we have [||M

12

A δτ Dnh||2 + ||M

12

B Dn+ 1

2h ||2 + ||∇×δτ E

nh||2 + ||∇× E

n+ 12

h ||2

+ε0µ0µλ2(||M−12

A δ2τ E

n+ 12

h ||2 + ||ωpM− 1

2A δτ E

nh||2 + ||ωpM

− 12

A En+ 1

2h ||2)

]1/2

≤ CT (τ2 + hp).


Nodal DG method: Hesthaven, Warburton’s 2008 bookOn triangular meshes, our finite element spaces are given by:

Uh = ψh ∈ L2(Ω) : ψh|K ∈ Pp, ∀ K ∈ Th, VVV h = (Uh)2.

Given proper initial approximations H0h ,D

− 12

h ,D− 3

2h , E

− 12

h ,E− 3

2h , for n ≥ 0 find

Dn+ 1

2h ,E

n+ 12

h ∈ V 0h, Hn+1

h ∈ Uh such that on any element Ki ∈ Th,∫Ki

δτDnh ·φh =

∫Ki

Hnh ·∇×φh + ∑

K∈νi

∫aik

φh×nik · Hnhik ,

ε0λ2

∫Ki

δ2τ E

n+ 12

h ·ϕh + ε0λ2

∫Ki

ω2p E

n+ 12

h ·ϕh

=∫

Ki

MAδ2τ D

n+ 12

h ·ϕh +∫

Ki

MBDn+ 1

2h ·ϕh,

µ0µ

∫Ki

δτHn+ 1

2h ψh =−

∫Ki

En+ 1

2h ·∇×ψh− ∑

K∈νi

∫aik

ψh ·nik ×En+ 1

2h ik ,

hold true for any φh,ϕh ∈ V0h,ψh ∈ Uh. En+ 1

2h ik denotes average of En+ 1

2h

on internal face aik , νi denotes all neighboring elements of K .Yunqing Huang (Xiangtan U.) July 18, 2019 29 / 69

PML

To simulate the cloak phenomenon, we surround the physical domainby a perfectly matched layer (PML), see Fig.8 (Right). In this paper, weuse the classical 2D Berenger PML, whose governing equations canbe written as

ε0∂EEE∂ t

+

(σy 00 σx

)EEE = ∇×Hz , (19)

µ0∂Hzx

∂ t+ σmxHzx =−

∂Ey

∂x, (20)

µ0∂Hzy

∂ t+ σmyHzy =

∂Ex

∂y, (21)

where Hz = Hzx + Hzy denotes the magnetic field, the parameters σiand σm,i , i = x ,y , are the electric and magnetic conductivities in the x-and y - directions, respectively. In our simulation, we use a PML with12 rectangular cells in thickness around the physical domain.


Numerical examples: hybrid edge elements

In our simulation, we choose

H1 = 0.05m,H2 = 0.2m,d = 0.2m,Ω = [−0.3,0.3] m× [0,0.3] m,

partition Ω by a uniform triangular mesh with h = 0.00625. The PMLregion surrounding Ω is partitioned by a uniform rectangular mesh.Real simulation is obtained with a final mesh with 53330 total edges,26960 triangles, 6258 rectangles,

τ = 2∗10−13s,Nt = 15000,T = 3.0ns.

Example 1. The plane wave source imposed at line x =−0.3:

Hz = 0.1sin(ωt),ω = 2πf , f = 3.0GHz.

The numerical magnetic fields Hz at different times are shown inFig.49.


Numerical examples: Ex 1

Figure 9: Ex1. The Hz fields at 5000, 7000, 10000, 15000 time steps.


Numerical examples: hybrid edge elements

Example 2. Gaussian incident wave imposed along a slanted liney = x + 0.45:

Hz(x ,y , t) = 0.1e−(y−0.15)2/(60L)2sin(ωt),

L = 0.004√

2, ω = 2πf , f = 6.0GHz.

The magnetic fields Hz at different time steps are presented in Fig.10.

To appreciate the cloak phenomenon, in Fig.11 we present themagnetic fields Hz obtained without the cloaking material. It is clearthat the cloak phenomenon disappears if the cloaking material isremoved.



Figure 10: Ex2. The Hz fields at 5000, 7000, 10000, 15000 time steps.



Figure 11: Ex2. The Hz fields at 5000, 7000, 10000, 15000 time stepsobtained with the cloaking material removed.






5 Major References


Arbitray star-shaped cloak: Yang, Li, Huang (SINUM2018)

Figure 12: The mesh used for the simulation of mushroom shaped cloak.Yunqing Huang (Xiangtan U.) July 18, 2019 37 / 69

A mushroom shaped cloak: Yang, Li, Huang (SINUM2018)

Figure 13: Electric fields Ey at various time steps. Top: t=6 ns, 12 ns, 15 ns;Bottom: t=24 ns, 30 ns, 45 ns.


Electromagnetic concentrator: Yang, Li, Huang, He(CiCP 2018)Coordinate transformation:

r ′ =

ab r , 0≤ r ≤ b,

c−ac−b r − b−a

c−b c, b ≤ r ≤ c,(22)

φ′ = φ , 0≤ φ ≤ 2π. (23)

Figure 14: Coordinate transformation for cylindrical concentrator: (Left) theoriginal space; (Right) the transformed space.


Electromagnetic concentrator: cont’d

Figure 15: Snapshots of electric field Ey : 2000 time steps (Top Left); 4000steps (Top Right); 8000 steps (Bottom Left); 10000 steps (Bottom Right).


Electromagnetic concentrator: cont’d

∂tD = ∇×H, (24)ε0εmax εr Ma∂ttE + ε0ω

2e εr MaE = ∂ttD + MbD, (25)

µ0µmax ∂ttH + µ0ω2mH =−∇×∂tE . (26)

εr = r ′+K1r ′ , where r ′ ∈ [a,c] and K1 = (b−a)c

c−b . The parametersεmax = c−b

c−a , ωe, µmax = b(c−b)a(c−a) and ωm come from the following Drude

models used to map εφ = r ′r ′+K1

and µ ′(r ′) =(

c−bc−a

)2 r ′+K1r ′ since they

can be less than one:

εφ = εmax −ω2

eω2 , µ

′(r ′) = µmax −ω2

mω2 .


Electromagnetic concentrator: stability

Theorem 8

Denote the energy

ENG(t) =[||∂tD||2 + ||M1/2

b D||2 + µ0µmax ||∂tH||2 + µ0||ωmH||2

+ε0εmax ||ε1/2r M1/2

a ∂tE ||2 + ε0||ωeε1/2r M1/2

a E ||2 + ||∇×∂tE ||2

+ε0µ0µmax

(εmax ||ε1/2

r M1/2a ∂ttE ||2 + ||ωeε

1/2r M1/2

a ∂tE ||2)]

(t).

Then for the solution of (24)-(26) satisfying the PEC boundarycondition (10) and any t ∈ [0,T ], there exists a constant C > 0 suchthat

ENG(t)≤ C ·ENG(0).


Electromagnetic rotator

r ′ = r , 0≤ r ≤ R2, (27)

θ′ =

θ + θ0, 0≤ r ≤ R1,

θ + θ0(R2−r)R2−R1

, R1 ≤ r ≤ R2,(28)

This transformation rotates the incoming wave by an angle θ0 in thecore region Ω1, and then gradually reduces the rotated angle θ0 to 0 inthe shell region Ω2.

θ0

R1

R2

Ω2

Ω1

Figure 16: Coordinate transformation of a cylindrical EM rotator.


Electromagnetic rotator: cont’d

Figure 17: Snapshots of electric fields Ex (Top) and Ey (Bottom) for thesimulation of the EM rotator: 12000 steps (left); 20000 steps (middle); 60000steps (right).






5 Major References


Superconvergence on Maxwell’s equations

• P. Monk: Numer Math 1992; NMPDE 1994.• Q. Lin, N. Yan: Gongcheng Shuxue Xuebao, 1996; Math.Comp.,1999; Q. Lin, J. Lin: Acta Math Sci Ser A Chin Ed, 2003; JCM2003; H. Xie, ACM 2008;• Q. Lin, J. Li: Math Comp 2008; Y. Huang, J. Li, Q. Lin: NMPDEs 2012(rectangular edge elements); Y. Huang, J. Li, et al: JCP 2011 (cubicedge elements); CMAME 2013 (linear triangular edge element); JSC2015 (linear tetrahedral edge element); CMAME 2018, JSC 2018 (2nd,3rd-order rectangular edge elements); L. Wang, Q. Zhang, Z. Zhang:JSC 2019 (arbitray order edge elements for cubics and rectangles).• E. Chung, Ciarlet, Yu, JCP 2013 (Staggered DG on Cartesian grids)• B. Cockburn (2016) LDG for time-harmonic Maxwell


2nd-order rectangular edge element

∇×∇×EEE + κ20EEE = fff in Ω⊂ R2, (29)

EEE ×nnn = 0 on ∂ Ω, (30)

Weak formulation: Find EEE ∈ H0(curl ;Ω) such that

a(EEE ,φφφ) := (∇×EEE ,∇×φφφ) + κ20 (EEE ,φφφ) = (f ,φφφ), ∀φφφ ∈ H0(curl ;Ω). (31)

Rectangle K = [xc−hx ,xc + hx ]× [yc−hy ,yc + hy ]. FE space VVV h:

VVV h = vvvh ∈ H0(curl ;Ω) : vvvh|K ∈Qk−1,k ×Qk ,k−1, ∀ K ∈Th. (32)

The FEM for solving (31): Find EEEh ∈VVV h such that

a(EEEh,φφφh) = (f ,φφφh), ∀φφφh ∈VVV h. (33)


Superconvergence for rectangular or cubic element

The following classic optimal estimate holds true:

||EEE −EEEh||L2 + ||∇× (EEE −EEEh)||L2 ≤ Chk , ∀ k ≥ 1.

• Y. Huang, J. Li, W. Yang, S. Sun: J. Comput. Phys. 2011 (3D, k=1)• Y. Huang, J. Li, Q. Lin: Numer. Methods Partial Differ. Equ. 2012(2D,k=1)• Y. Huang, J. Li, C. Wu: Comput. Methods Appl. Mech. Engrg. 2018(k=2, 3)• L. Wang, Q. Zhang, Z. Zhang, J Sci Comput 2019 (any k)


Nedelec interpolation operator Πh

For any vvv ∈ H(curl ,Ω), (Πhvvv)|K = ΠKvvv ∈Qk−1,k ×Qk ,k−1 satisfies∫li(vvv −ΠKvvv) · ttt iqdl = 0, ∀q ∈ Pk−1(li), i = 1, · · · ,4, (34)∫

K(vvv −ΠKvvv) ·qqqdxdy = 0, ∀qqq ∈Qk−1,k−2×Qk−2,k−1. (35)

where li : edges of element K , ttt i : unit tangent vectors along edge li .

Figure 18: denotes the 2nd order Gauss points of element K .Yunqing Huang (Xiangtan U.) July 18, 2019 49 / 69

k=2, The basis functions: cont’d

Lemma 9The curl conforming hierarchical basis functions in Q1,2×Q2,1 are:

φφφ112 =

[1

hx(1− y−yc

hy)

0

], φφφ

212 =

[1

hxx−xc

hx(1− y−yc

hy)

0

], φφφ

1K =

[1

hx[( y−yc

hy)2−1]

0

],

φφφ123 =

[0

1hy

(1 + x−xchx

)

], φφφ

223 =

[0

1hy

y−ycl2

(1 + x−xchx

)

], φφφ

2K =

[0

1hy

[( x−xchx

)2−1]

],

φφφ134 =

[1

hx(1 + y−yc

hy)

0

], φφφ

234 =

[1

hxx−xc

xx(1 + y−yc

hy)

0

], φφφ

3K =

[1

hxx−xc

hx[( y−yc

hy)2−1]

0

],

φφφ114 =

[0

1hy

(1− x−xchx

)

], φφφ

214 =

[0

1hy

y−ychy

(1− x−xchy

)

], φφφ

4K =

[0

1hy

y−ychy

[( x−xchx

)2−1]

].


k=2, The basis functions: cont’d

Lemma 10

For any function uuu ∈ H(curl ;K ), interpolation ΠSK u can be written as

ΠSK uuu =

2

∑i=1

(c i12φφφ

i12 + c i

23φφφi23 + c i

34φφφi34 + c i

14φφφi14) +

4

∑j=1

c jK φφφ

jK := GGG +

4

∑j=1

c jK φφφ

jK , (36)

c112 =

14

∫e12

uuu · ttt12ds, c212 =

∫e12

(uuu · ttt12) · 34hx

(x−xc)ds, ttt12 = (1,0)′,

c123 =

14

∫e23


∫e23

(uuu · ttt23) · 34hy

(x−xc)ds, ttt23 = (0,1)′,

c134 =

14

∫e34


∫e34

(uuu · ttt34) · 34hx

(x−xc)ds, ttt34 = (1,0)′,

c114 =

14

∫e14


∫e14

(uuu · ttt14) · 34hy

(x−xc)ds, ttt14 = (0,1)′,

c1K =

∫K

(uuu−GGG) ·qqq1dxdy , c3K =

∫K

(uuu−GGG) ·qqq3dxdy ,

c2K =

∫K

(uuu−GGG) ·qqq2dxdy , c4K =

∫K

(uuu−GGG) ·qqq4dxdy ,

qqq1 =

[−38hy0

], qqq3 =

[−98hy

x−xchx

0

], qqq2 =

[0−38hx

], qqq4 =

[0

−98hx

y−ychy

].


Superconvergent interpolation errors at Gauss points

Theorem 11

For a rectangle K = [xc−hx ,xc + hx ]× [yc−hy ,yc + hy ], denote theGauss points

P1(xc−1√3

hx ,yc−1√3

hy ), P2(xc +1√3

hx ,yc−1√3

hy ),

P3(xc−1√3

hx ,yc +1√3

hy ), P4(xc +1√3

hx ,yc +1√3

hy ).

Then for any function u ∈ H(curl ;Ω), the Nedelec interpolationΠS

K u ∈Q1,2×Q2,1 satisfies the estimates:

(i) |(u−ΠSK u)(Pi)| ≤ Ch3

K , ∀ u ∈ (C3(K ))2, i = 1,2,3,4, (37)

(ii) |∇× (u−ΠSK u)(Pi)| ≤ Ch3

K , ∀ u ∈ (C4(K ))2, i = 1,2,3,4. (38)


Techniques: integral identity + superclose

Huang, Li, Lin (NMPDE 2012):

Lemma 12

For any uuu ∈ (Hk+1(K ))2,we have:∫K

(uuu−ΠKuuu) ·vvvdxdy = O(hk+1K )‖uuu‖k+1,K‖vvv‖0,K , ∀vvv ∈VVV h. (39)

Furthermore, for any uuu ∈ (Hk+2(K ))2, we have:∫K

∇× (uuu−ΠKuuu) ·ψdxdy = 0, ∀ψ ∈ Wh = ∇×VVV h. (40)

Theorem 13

Let EEE and EEEh be solutions of (31) and (33), then we have:

‖ΠhEEE −EEEh‖0 +‖∇× (ΠhEEE −EEEh)‖0 ≤ Chk+1‖EEE‖Hk+1(Ω). (41)


Superconvergence in discrete norms

For rectangular (and cubic) edge elements Qk−1,k ×Qk ,k−1,k ≥ 1,define discrete l2 norms:

‖uuu‖l2 :=(

∑K∈Th

k

∑i ,j=1

(ωi ωj |uuu(pij )|2

)·hx hy

) 12, ωi : weights, pij : Gauss points.

Theorem 14Let (EEE ,H) and (EEEh,Hh) be solutions of (31) and (33), then we have

‖EEE −EEEh‖l2 +‖curl(EEE −EEEh)‖l2 ≤ Chk+1, ∀ EEEh ∈VVV h with k ≥ 1. (42)


Numerical results

To justify our theoretical analysis, we use the rectangular edgeelements of k = 2 and k = 3 to solve the model problem (33) with

κ0 = 1 and exact solution EEE =

[cos(πx)sin(πy)−sin(πx)cos(πy)

]. Note that this

solution satisfies the PEC boundary condition (30) and the divergence

free condition ∇ ·EEE = 0. Define ‖uuu‖l2∗∗ :=(

∑K∈Th|uuu(xc ,yc)|2 ·4hxhy

) 12.

Figure 19: A 64×64 nonuniform mesh.


Numerical results for k = 2

Table 1: Convergence rates on the 2nd nonuniform mesh with k = 2

mesh size ||uuu−uuuh||0 order ||∇× (uuu−uuuh)||0 order4×4 0.0247221 0.1518518×8 0.0072984 1.7601 0.0456619 1.7336

16×16 0.00187582 1.9601 0.0117735 1.955532×32 0.000471935 1.9908 0.00296446 1.989764×64 0.000118167 1.9978 0.000742413 1.9975

Table 2: Superconvergence on the 2nd nonuniform mesh with k = 2

mesh size ||uuu−uuuh||l2 order ||∇× (uuu−uuuh)||l2 order4×4 0.00828564 0.03006938×8 0.00103097 3.0066 0.00361955 3.0544

16×16 0.000129557 2.9924 0.000449665 3.008932×32 1.62237e-05 2.9974 5.61308e-05 3.002064×64 2.02893e-06 2.9993 7.014e-06 3.0005


Numerical results for k = 2,3: cont’d

Table 3: Convergence rates at element centers on the 2nd piecewise uniformmesh with k = 2

mesh size ||uuu−uuuh||l2∗∗ order ||∇× (uuu−uuuh)||l2∗∗ order4×4 0.020146 0.1740058×8 0.00581348 1.7930 0.0515267 1.7557

16×16 0.00148054 1.9733 0.0132886 1.955132×32 0.000371572 1.9944 0.00334654 1.989464×64 9.2979e-05 1.9987 0.000838144 1.9974

Table 4: Superconvergence on the 2nd piecewise uniform mesh with k = 3

mesh size ||u−uh||l2∗ order ||∇× (u−uh)||l2∗ order4×4 0.0256136 0.007960598×8 0.00195216 3.7138 0.000616283 3.6912

16×16 0.000126117 3.9522 3.997e-05 3.946632×32 7.94294e-06 3.9889 2.51977e-06 3.987664×64 4.97367e-07 3.9973 1.5782e-07 3.9969


Superconvergence of tetrahedra edge element

• In 1992, Monk (Numer. Math. 1992) solved the time-harmonicMaxwell equations using Nede1ec’s first type linear tetrahedralelements, and showed that the numerical solutions gave O(h2)convergence in the discrete maximum norm:

||E −Eh||∞,h = maxxe∈E|(E(xe)−Eh(xe)) · te|,

which indicates that type one linear tetrahedral edge elements havesuperconvergence for the solutions at the midpoints of interior edges.•We filled this gap by providing a theoretical justification of thissuperconvergence phenomenon in 2015 (Huang, Li, Wu, Yang: J SciComput 2015)


Superconvergence for Nedelec interpolation

Assume that Ω is first partitioned into cubics, then each cube is furtherpartitioned into six tetrahedra as shown in Fig. 20. Note that midpointO(xo,yo,zo) of edge A1A7 is shared by the following six tetrahedra:

K1 = K1732,K2 = K1758,K3 = K1726,K4 = K1784,K5 = K1765,K6 = K1743.

Figure 20: The edge A1A7 shared by six tetrahedra.Yunqing Huang (Xiangtan U.) July 18, 2019 59 / 69

Superconvergence for Nedelec interpolation: cont’d

Theorem 1

Let ω1 = K1∪K2. Then for any uuu ∈ C2(ω1), we have

(uuu− 12

(Π1huuu|K1 + Π1

huuu|K2))(O) = O(h2).

Theorem 2

Let ω1 = K1∪K2 and φh = φ117. For any uuu ∈ C2(ω1) we have∫

ω1

(uuu−Π1huuu) ·φh dK ≤ Ch2||∂ 2uuu||∞||φh||L1(ω1). (43)



Similar results as Theorems 1 and 2 hold true for a pair of tetrahedraformed by K3 = K1726 and K4 = K1784, and K5 = K1765 and K6 = K1743,respectively. More precisely, we have

(uuu− 12

(Π1huuu|K3 + Π1

huuu|K4))(O)≤ Ch2, (44)

(uuu− 12

(Π1huuu|K5 + Π1

huuu|K6))(O)≤ Ch2, (45)

and ∫K3∪K4

(uuu−Π1huuu) ·φ1

17 dK ≤ Ch2||∂ 2uuu||∞||φ117||L1(K3∪K4), (46)∫

K5∪K6

(uuu−Π1huuu) ·φ1

17 dK ≤ Ch2||∂ 2uuu||∞||φ117||L1(K5∪K6). (47)



The results (44)–(45) hold true for the edge midpoints between twocubics such as the midpoint of edge A1A6 (cf. Figure 21) shared by apair of tetrahedra such as K5 and K7, and K3 and K8; and the midpointof edge A5A6 (cf. Figure 21) shared by a pair of tetrahedra such as K5and K9, K8 and K12, and K10 and K11.

Figure 21: (Left) The diagonal edge A1A6 of the common face between twocubics; (Right) The edge A5A6 shared by six tetrahedra..Yunqing Huang (Xiangtan U.) July 18, 2019 62 / 69

Superconvergence for Maxwell equations

Consider the time-harmonic Maxwell equations

∇×∇×E −k20 E = f , in Ω, (48)

subject to PEC BC. Here k20 = ω

√εµ is the wave number.

A standard finite element method with edge elements: Find Eh ∈ V 1h

such thata(Eh,φh) = (f ,φh), ∀ φh ∈ V 1

h . (49)


Superconvergence for Maxwell equations: cont’d

Theorem 3

Assume that the domain Ω is covered by a regular cubic mesh, whereeach cubic element is further divided into six tetrahedra (cf. Figures20 and 21). Let E and Eh be the solutions of (48) and (49),respectively. If E ∈ C2(Ω), then for sufficiently small h we have

||Π1hE −Eh||H(curl;Ω) ≤ C(h2||∂ 2E ||∞ + hδ+ 1

2 ||E −Eh||H(curl;Ω)),

where parameter δ ∈ (0, 12 ] is the same as Lemma 7.7 of Monk’s book.


Superconvergence for Maxwell equations: cont’d

Figure 22: Meshes with(L)/without (R) superconvergence

R(uuu)(O) =1

#(ωK ) ∑∂K∈ωK

uuu|K (O), |u|l2 ,Ω =

(1

Ne∑

K∈Th

∑xe∈∂K

|u(xe)|2)1/2

.

Theorem 4For a sufficiently smooth solution E of (48), we have

||E −R(Eh)||l2,Ω ≤ Ch32 +δ . (50)


Numerical test on tetrahedra






5 Major References


Major References

• Li and Huang, Time-Domain Finite Element Methods for Maxwell’sEquations in Metamaterials, Springer Series in ComputationalMathematics, vol.43, Springer, 2013.

• Li, Huang, Yang, Wood, SIAM J. Appl. Math. 74(4) (2014) 1136-1151• Li, Huang, Yang, Math. Comp. 84 (2015), no. 292, 543-562• Yang, Li, Huang, SIAM J. Numer. Anal. 56(1) (2018) 136-159• Y. Huang, J. Li, W. Yang, S. Sun: J. Comput. Phys. 2011• Y. Huang, J. Li, Q. Lin: Numer. Methods Partial Differ. Equ. 2012• Y. Huang, J. Li, C. Wu: Comput. Methods Appl. Mech. Engrg. 2018


Thank you for your attention!


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