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Non Spurious Mixed Spectral Element Methods for
Maxwell’s Equations
Gary Cohen, Alexandre Sinding
To cite this version:
Gary Cohen, Alexandre Sinding. Non Spurious Mixed Spectral Element Methods for Maxwell’s
Equa-tions. The 9th International Conference on Mathematical and Numerical Aspects of Waves
Propaga-tion, Jun 2009, Pau, France. �hal-01956669�
G. Cohen1 and A. Sinding1
INRIA, Domaine de Voluceau, Rocquencourt - BP 105, 78153 Le Chesnay Cedex, France, email: gary.cohen@inria.fr
1
Introduction
Solving Maxwell’s equations in time domain by finite element methods (FEM) is a challenging problem from two points of view: First, one must overcome the difficult problem of inverting the mass matrix (produced by FEM) at each time-step. Secondly, one must get a good approxima-tion of the non empty kernel of the curl operator which generates spurious waves when not well approximated.
The first point is generally solved by using a mass-lumping technique, not obvious for tri-angular or tetrahedral meshes [10,11] but easier and more efficient for quadrangular and hexa-hedral meshes. These last approaches are called spectral element methods and are particularly efficient in terms of storage and computational time in their mixed formulations described in [4,6,8].
The second point was partially overcomed until now. A simple way to solve this problem is to use the first family of edge elements described in [14] which ensure a good approximation of the curl’s kernel. Unfortunately, such elements pro-vide mass-lumping only on orthogonal meshes [7]. For discontinuous Galerkin methods (DGM), spurious modes become evanescent by using a dissipative jump term as shown in [5,12,13]. An-other kind of jump, based on the normal compo-nent of a H(curl) field was used in [5] for the sec-ond family of edge elements [15]. Unfortunately, although efficient for convex domains, this term does not provide a correct model for singulari-ties produced by reentrant corners, as we show in this paper. For continuous elements, the problem was solved by adding a divergence penalty term, which leads to a substantial additional cost and demanded a lot of work for modeling reentrant corners [1,3,9].
In this paper, we describe a new continu-ous approximation of Maxwell’s equations well-suited to mass-lumping and which ensures low-storage. Then, we introduce a dissipative jump derived from DGM to get rid of spurious waves for both edge and continuous elements. This new approach leads to efficient spectral elements for Maxwell’s equations which are cheaper than
DGM. On the other hand, this approach pro-vides a good approximation of singularities gen-erated by reentrant corners.
2
The Continuous Model
We want to solve Maxwell’s equations in inho-mogenous anisotropic lossy medium. This model reads
Find (E, H): Ω×]0, T [→ IR3 such that
ε∂E
∂t − ∇ × H + σE = −J, (1)
µ∂H
∂t + ∇ × E = 0, (2)
where Ω ⊂ IR3
, J is a given function of time and space. ε, σ and µ are the tensors of permittivity,
conductivity and permeability which read ε I3,
σ I3 and µ I3, where I3is the identity matrix, in
the isotropic case.
To these equations, we add homogeneous ini-tial conditions on (E, H) and the perfectly con-ducting condition E × n = 0 on ∂Ω.
The variational formulation of (1)-(2) can be written as Find E ∈ L2(0, T ; H 0(curl, Ω)) and H ∈ L2(0, T ;L2(Ω)3) such that d dt Z Ω εE· ϕ dx − Z Ω H· ∇ × ϕ dx + Z Ω σE· ϕ dx = − Z Ω J· ϕ dx, ∀ϕ ∈ H0(curl, Ω), (3) d dt Z Ω µH· ψ dx + Z Ω ∇ × E · ψ dx = 0, ∀ψ ∈L2(Ω)3 , (4)
We recall the definition of H0(curl, Ω):
H0(curl, Ω) = n ϕsuch that ϕ ∈L2(Ω)3 , ∇ × ϕ ∈L2(Ω)3 , ϕ× n = 0o
where n is the outward unit normal of ∂Ω. Both approximations by edge or continuous elements can be derived from (3)-(4) but not dis-continuous Galerkin methods.
2 G. Cohen and A. Sinding
The continuous model being defined, we can now describe the different kinds of approxima-tions treated in this paper.
3
A General Framework for
Spectral Elements
A first step for constructing mixed spectral ele-ment approximations is to define the
interpola-tion points and funcinterpola-tions on the unit cube bKas
follows:
Let 1 ≤ i ≤ r + 1, bϕi(ˆx) a Lagrange
in-terpolation polynomial of order r and nξˆℓ
or+1
ℓ=1
the Gauss-Lobatto quadrature points on interval [0, 1]. These points are the interpolation points
for bϕi, i.e. bϕi( ˆξℓ) = δiℓ, where δiℓ is the
Kro-necker symbol.
Now, we define the interpolation functions
on bKas products of the above defined
interpola-tion funcinterpola-tions. The corresponding interpolainterpola-tion points are the tensor products of the 1D
Gauss-Lobatto points, namely bξℓ,m,n=
ˆ
ξℓ, ˆξm, ˆξn
.
So, we get bϕi,j,k(bξℓ,m,n) = bϕi( ˆξℓ) bϕj( ˆξm)
b
ϕk( ˆξn) = δiℓδjmδkn. Of course, bϕi,j,k ∈ Qr,
where Qr is the polynomial space defined as
Qr= r X i=0 r X j=0 r X k=0 ai,j,kxi1x j 2x k 3, ai,j,k∈ IR
Now, at each point bξi,j,k, we have three
basis functions nϕbs i,j,k o3 s=1 such that bϕ s i,j,k = b
ϕi,j,kes, where es is a unit basis vector of IR3.
Let Fp the transform such that Fp( bK) =
Kp, where Kp is an hexahedron of a mesh M.
In a second step, except for continuous elements,
we define the basis functions on Kp as follows:
ϕsp,i,j,k◦ Fp= DFp∗−1ϕbsi,j,k, (5)
where DF∗−1
p is the transpose inverse of the
Ja-cobian matrix of Fp.
DFp∗−1 is the H(curl)-conforming mapping,
i.e. this mapping keeps the H(curl) character of edge elements. Although useful for edge elements only, this definition will be used for all kinds of mixed spectral approximations for the following reason: for two basis functions defined as in (5), we have Z Kp ∇ × ϕsp,i,j,k· ϕs ′ p,i′,j′,k′dx = sgn(Jp) Z b K b ∇ × bϕsi,j,k· bϕsi′′,j′,k′dˆx, (6)
where Jp = det(DFp) is the Jacobian of DFp
and b∇ is the gradient with respect to the ˆx
co-ordinates.
In other words, the the knowledge of the stiff-ness integrals of the basis functions on the unit
cube and the signe of the Jacobian on Kpenable
the knowledge of the stiffness integrals all over the mesh, which induces a huge gain of storage. With these definitions and by computing all the integrals by the mean of Gauss-Lobatto quadrature rules, we get block-diagonal mass matrices and very sparse stiffness matrices which provide a low-storage and fast algorithm, as de-scribed in details in [4].
These definitions can be used for Discontinu-ous Galerkin Methods or edge elements [3]. The only difference will lean on the continuity prop-erties of the elements.
4
Spectral Continuous Elements
4.1 Classical Approximation
Continuous finite elements for Maxwell’s equa-tions were first introduced on the second-order (curl-curl) equation. This approximation gener-ates, of course, spurious waves due to the bad approximation of the divergence of the electric field. The idea to get rid of these waves is based on the well-known relation
∇ × (∇E) = ∇(∇ · E) − ∆E. (7)
In the vacuum or in the air (i.e. ε = ε I3, µ = µ I3
and σ = 0), we have ∇ × (∇E) = −∆E since ∇ · E = 0. So, in order to get rid of the spurious waves, one adds a penalty term in ∇(∇·E). The discrete formulation using the following sub-space of H(curl) Ur h= {vh∈ H1 0(Ω) 3 such that ∀Kj∈ M, vh|Kj◦ Fj∈ (Qr) 3 }. (8) reads d2 dt2 Z Ω ε Eh· ϕhdx − Z Ω 1 µ∇ × Eh· ∇ × ϕhdx −ρ Z Ω ∇ · Eh∇ · ϕhdx = − Z Ω J · ϕhdx, ∀ϕh∈ Urh, (9) where ρ ∈ IR+∗.
As we said in our introduction, this approach presents a double drawback: the penalty term is expensive and we still get spurious modes for reentrant corners. For these reasons, we con-struct a new continuous element approximation in the next section.
4.2 Mixed Spectral Continuous Elements
The idea of our approximation is to take E ∈ Ur
h
and H ∈ Wr
hand to derive the approximate
for-mulation from (3)-(4). Moreover, as in the two previous approximations, we add a jump term to the second equation which is written in a dis-continuous space. We get
Find Eh ∈ L2(0, T ; Urh) and Hh ∈ L2(0, T ; Wr h) such that d dt Z Ω εEh· ϕhdx − Z Ω Hh· ∇ × ϕhdx + Z Ω σEh· ϕhdx = − Z Ω J· ϕhdx, ∀ϕh∈ Urh, (10) X Kj∈M ( d dt Z Kj µHh· ψhdx + Z Kj ∇ × Eh· ψhdx + δ′′ Z ∂Kj [[n × Hh]] Kj ∂Kj · (n × ψh) ds ) = 0, ∀ψh∈ Wrh. (11) where δ′′∈ IR+∗.
One can easily derive the energy identity dE dt(t) = − X Ki∩Kj δ′′k[[H × n]]Ki∩Kjk 2 Γ,
which shows that our penalty term is dissipative. Now, an important algorithmic issue must be considered for this method. Actually, we don’t have the good property given in (6) since
Z Kj ∇ × Eh· ψhdx = Z b K |Jj| (∇ × Eh) ◦ Fj· ψh◦ Fjdˆx = Z b K |Jj| (DFj∗−1∇) × bb Eh· DFj∗−1ψbhdˆx. (12) (12) shows that, because of the definition of
Urh, the local character of the stiffness integral
is lost. That should imply of important stiffness matrix. For this reason, we have to transform the curl term in (12) as follows
(DFj∗−1∇) × bb Eh = (DFj∗−1∇) × (DFb j∗−1DFj∗Ebh) =DFj Jj b ∇ × (DFj∗Ebh) (13) From (13), we get Z Kj ∇ × Eh· ψhdx = sgn(Jj) Z b K b ∇ × (DFj∗Ebh) · bψhdˆx. (14)
(14) can be decomposed into Z b Kb vh· bϕhdˆx= Z b K (DF∗ jEbh) · bϕhdˆx, b vh∈ Wrh, ϕbh∈ Wrh, (15) Z Kj ∇ × Eh· ψhdx = sgn(Jj) Z b K b ∇ × bvh· bψhdˆx. (16)
In terms of matrices, (15) leads to a simple
product by a block mass matrix Mh and (16) a
locally defined stiffness matrix similar to those used in the two previous methods.
The stiffness integral of (10) can be treated in the same manner:
Z Ω Hh· ∇ × ϕhdx = X Kj∈M Z Kj Hh· ∇ × ϕhdx = X Kj∈M Z b K |Jj| Hh◦ Fj· (∇ × ϕh) ◦ Fjdˆx = X Kj∈M sgn(Jj) Z b K c Hh· b∇ × (DFj∗ϕbh) dˆx (17) (14) and (17) show that the matrices derived from these integrals are transposed one to the other.
Now, by taking basis functions as defined in section 3 and integrating by using a Gauss-Lobatto rule, we get the following discrete prob-lem:
DhE− Mh∗R
∗
hH+ ΣhE+ eDhJ= 0, (18)
BhH− RhMhE+ δ′′ShH = 0, (19)
where E and H are the vectors of thes degrees of freedoms for the approximate fields, and
• Dh, eDh and Σhare diagonal matrices,
• Bh is a 3 × 3 symmetric block-diagonal
ma-trix,
• Mh is a mass matrix derived from the
Jaco-bian matrices,
• Rh is a sparse stiffness matrix which need
a storage on the unit cube only (when com-puted element by element),
4 G. Cohen and A. Sinding
After a leapfrog discretization in time, (18)-(19) is actually solved as follows
Wn+1 2 = R∗hHn+12, Dh En+1− En ∆t − M ∗ hWn+ 1 2 +Σh En+1+ En 2 + eDhJ n+1 2 = 0, Vn = M hEn Bh Hn+12 − Hn−12 ∆t − RhV n +δ′′S h Hn+1 2+n−12 2 = 0. (20)
where V and W are auxiliary variable.
(20)-(20) provide a low storage and fast al-gorithm.
The approximation being defined, we must now study its properties.
4.3 Eigenvalue Analysis
In order to test the efficiency of the jump term to get rid of the spurious modes, we compute the eigenmodes of the time-harmonic problem. in Fig. , we show that all the spurious modes are suppressed by adding the jump term. In fact, the spurious modes are shifted to the complex plane with the same sign of the imaginary part, which produces evanescent waves. This term also works for reentrant corners.
0 10 20 30 40 50 60 70 80 90 100 0 20 40 60 80 100 120 140 0 10 20 30 40 50 60 70 80 90 100 0 20 40 60 80 100 120 140 160 180
Fig. 1.Eigenmodes without dissipation (above) and with dissipation (below)
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