The Curious Importance of
Function Spaces
for the Maxwell Equations
Martin Costabel
IRMAR, Université de Rennes 1
WAVES 2011
Simon Fraser University Vancouver, 24 – 29 July 2011
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
1 / 48
1
Are function spaces important?
What google has to say
The Dirichlet problem
2
Regularized Variational Formulation of Maxwell Equations
The Maxwell eigenvalue problem
Regularized variational formulations
Numerical observations
Explanation
Weighted Regularization
Numerical evidence for WRM
3
Regularized Boundary Integral Equations for Maxwell Equations
The Electrical Field Integral Equation
The Regularized EFIE
4
Lippmann-Schwinger Equation for the Magnetic Scattering Problem
Electromagnetic Transmission Problems
The Volume Integral Equation
The dielectric problem
The magnetic problem
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
2 / 48
From the WEB. Google search for “function spaces are important”
Function spaces are important and natural examples of
abstract Banach lattices
http://eom.springer.de
However. . .
Introducing Function Space Yielding
Are you already optimizing your revenues and profits on your
function space? The workshop will show you practical techniques how
to calculate the price you should quote, not leaving any money on the table.
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
3 / 48
From the WEB. Google search for “function spaces are important”
Function spaces are important and natural examples of
abstract Banach lattices
http://eom.springer.de
However. . .
Introducing Function Space Yielding
Are you already optimizing your revenues and profits on your
function space? The workshop will show you practical techniques how
to calculate the price you should quote, not leaving any money on the table.
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
3 / 48
From the WEB. Google search for “function spaces are important”
Function spaces are important and natural examples of
abstract Banach lattices
http://eom.springer.de
However. . .
Introducing Function Space Yielding
Are you already optimizing your revenues and profits on your
function space? The workshop will show you practical techniques how
to calculate the price you should quote, not leaving any money on the table.
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
3 / 48
From the WEB. Google search for “function spaces are important”
Function spaces are important and natural examples of
abstract Banach lattices
http://eom.springer.de
However. . .
Introducing Function Space Yielding
Are you already optimizing your revenues and profits on your
function space? The workshop will show you practical techniques how
to calculate the price you should quote, not leaving any money on the table.
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
3 / 48
spaces
are important and natural examples of
From the WEB. Google search for “function spaces are important”
anach lattices
http://eom.springer.de
..
Function spaces are important and natural examples of
abstract Banach lattices
http://eom.springer.de
However. . .
Introducing Function Space Yielding
Introducing Function Space Yielding
ready optimizing your revenues and profits on your
Are you already optimizing your revenues and profits on your
pace?
The space?
workshop
will show you practical techniques how
function
The workshop will show you practical techniques how
the price
you should
notquote,
leaving
any money
ononthe
to calculate
the pricequote,
you should
not leaving
any money
the table.
table.
http://www.xotels.com/conference-banquet-and-group-revenue-management
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
3 / 48
spaces
are important and natural examples of
From the WEB. Google search for “function spaces are important”
anach lattices
http://eom.springer.de
..
Function spaces are important and natural examples of
abstract Banach lattices
http://eom.springer.de
However. . .
Introducing Function Space Yielding
Introducing Function Space Yielding
ready optimizing your revenues and profits on your
Are you already optimizing your revenues and profits on your
pace?
The space?
workshop
will show you practical techniques how
function
The workshop will show you practical techniques how
the price
you should
notquote,
leaving
any money
ononthe
to calculate
the pricequote,
you should
not leaving
any money
the table.
table.
http://www.xotels.com/conference-banquet-and-group-revenue-management
Somewhat more seriously. . .
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
3 / 48
The Dirichlet problem
Ω: bounded polygon or polyhedron
(Dir)
∆u = f
in Ω,
u = 0 on ∂ Ω
One can consider (Dir) in
Sobolev spaces
H s (Ω), Wpm (Ω)
Besov spaces
Bps,q (Ω)
Hölder spaces
C α (Ω)
m,~β ,~δ
with or without weight
etc.
Wp
(Ω)
In some of these spaces
there is no existence result (Example: H 2 (Ω), nonconvex Ω)
there is no uniqueness result (Example: L2 (Ω), nonconvex Ω)
There is, however, a sanity result...
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
4 / 48
The Dirichlet problem
Ω: bounded polygon or polyhedron
(Dir)
∆u = f
in Ω,
u = 0 on ∂ Ω
One can consider (Dir) in
Sobolev spaces
H s (Ω), Wpm (Ω)
Besov spaces
Bps,q (Ω)
Hölder spaces
C α (Ω)
m,~β ,~δ
with or without weight
etc.
Wp
(Ω)
In some of these spaces
there is no existence result (Example: H 2 (Ω), nonconvex Ω)
there is no uniqueness result (Example: L2 (Ω), nonconvex Ω)
There is, however, a sanity result...
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
4 / 48
The Dirichlet problem
Observation
Let X1 and X2 be two “reasonable” function spaces on Ω from the
afore-mentioned list.
Let f ∈ ∆X1 ∩ ∆X2 and let u1 ∈ X1 , u2 ∈ X2 both be solutions of (Dir).
If X1 and X2 are such that in both spaces the homogeneous Dirichlet
problem has only the trivial solution,
then
=⇒
u1 = u2 .
Function spaces are
useful: description of singularities, stability, error estimates, . . . ,
but not really important.
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
5 / 48
The Dirichlet problem
Observation
Let X1 and X2 be two “reasonable” function spaces on Ω from the
afore-mentioned list.
Let f ∈ ∆X1 ∩ ∆X2 and let u1 ∈ X1 , u2 ∈ X2 both be solutions of (Dir).
If X1 and X2 are such that in both spaces the homogeneous Dirichlet
problem has only the trivial solution,
then
=⇒
u1 = u2 .
Function spaces are
useful: description of singularities, stability, error estimates, . . . ,
but not really important.
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
5 / 48
Outline
1
Are function spaces important?
What google has to say
The Dirichlet problem
2
Regularized Variational Formulation of Maxwell Equations
The Maxwell eigenvalue problem
Regularized variational formulations
Numerical observations
Explanation
Weighted Regularization
Numerical evidence for WRM
3
Regularized Boundary Integral Equations for Maxwell Equations
The Electrical Field Integral Equation
The Regularized EFIE
4
Lippmann-Schwinger Equation for the Magnetic Scattering Problem
Electromagnetic Transmission Problems
The Volume Integral Equation
The dielectric problem
The magnetic problem
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
6 / 48
Outline
1
Are function spaces important?
What google has to say
The Dirichlet problem
2
Regularized Variational Formulation of Maxwell Equations
The Maxwell eigenvalue problem
Regularized variational formulations
Numerical observations
Explanation
Weighted Regularization
Numerical evidence for WRM
3
Regularized Boundary Integral Equations for Maxwell Equations
The Electrical Field Integral Equation
The Regularized EFIE
4
Lippmann-Schwinger Equation for the Magnetic Scattering Problem
Electromagnetic Transmission Problems
The Volume Integral Equation
The dielectric problem
The magnetic problem
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
6 / 48
Outline
1
Are function spaces important?
What google has to say
The Dirichlet problem
2
Regularized Variational Formulation of Maxwell Equations
The Maxwell eigenvalue problem
Regularized variational formulations
Numerical observations
Explanation
Weighted Regularization
Numerical evidence for WRM
3
Regularized Boundary Integral Equations for Maxwell Equations
The Electrical Field Integral Equation
The Regularized EFIE
4
Lippmann-Schwinger Equation for the Magnetic Scattering Problem
Electromagnetic Transmission Problems
The Volume Integral Equation
The dielectric problem
The magnetic problem
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
6 / 48
Outline
1
Are function spaces important?
What google has to say
The Dirichlet problem
2
Regularized Variational Formulation of Maxwell Equations
The Maxwell eigenvalue problem
Regularized variational formulations
Numerical observations
Explanation
Weighted Regularization
Numerical evidence for WRM
3
Regularized Boundary Integral Equations for Maxwell Equations
The Electrical Field Integral Equation
The Regularized EFIE
4
Lippmann-Schwinger Equation for the Magnetic Scattering Problem
Electromagnetic Transmission Problems
The Volume Integral Equation
The dielectric problem
The magnetic problem
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
6 / 48
1
Are function spaces important?
What google has to say
The Dirichlet problem
2
Regularized Variational Formulation of Maxwell Equations
The Maxwell eigenvalue problem
Regularized variational formulations
Numerical observations
Explanation
Weighted Regularization
Numerical evidence for WRM
3
Regularized Boundary Integral Equations for Maxwell Equations
The Electrical Field Integral Equation
The Regularized EFIE
4
Lippmann-Schwinger Equation for the Magnetic Scattering Problem
Electromagnetic Transmission Problems
The Volume Integral Equation
The dielectric problem
The magnetic problem
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
7 / 48
Time-harmonic Maxwell equations
curl E = i ω µ H
curl H = −i ωε E + J
In this section: Domain Ω ⊂ R3 , ε = µ = 1, J = 0.
The condition div E = div H = 0 follows if ω 6= 0.
E ×n = 0
& H ·n = 0
on
∂Ω
Eigenfrequencies of a cavity with perfectly conducting walls.
Second order system for E:
curl curl E − ω 2 E = 0
Simplest variational formulation
Find ω 6= 0, E ∈ H0 (curl, Ω) \ {0} such that
Z
Z
curl E · curl F = ω 2 E · F
∀F ∈ H0 (curl, Ω) :
Ω
Ω
Energy space: H0 (curl, Ω) = {u ∈ L2 (Ω) | curl u ∈ L2 (Ω); u × n = 0}
= closure in H (curl, Ω) of C0∞ (Ω)3
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
8 / 48
Time-harmonic Maxwell equations
curl E = i ω µ H
curl H = −i ωε E + J
In this section: Domain Ω ⊂ R3 , ε = µ = 1, J = 0.
The condition div E = div H = 0 follows if ω 6= 0.
E ×n = 0
& H ·n = 0
on
∂Ω
Eigenfrequencies of a cavity with perfectly conducting walls.
Second order system for E:
curl curl E − ω 2 E = 0
Simplest variational formulation
Find ω 6= 0, E ∈ H0 (curl, Ω) \ {0} such that
Z
Z
curl E · curl F = ω 2 E · F
∀F ∈ H0 (curl, Ω) :
Ω
Ω
Energy space: H0 (curl, Ω) = {u ∈ L2 (Ω) | curl u ∈ L2 (Ω); u × n = 0}
= closure in H (curl, Ω) of C0∞ (Ω)3
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
8 / 48
Regularized formulation
Simple variational formulation
E ∈ H0 (curl, Ω)\{0} : ∀F ∈ H0 (curl, Ω) :
Z
curl E · curl F = ω 2
Z
E ·F
Ω
Ω
Galerkin discretization:
Restriction to finite-dimensional subspace Vh , h → 0.
Good: Eigenfrequencies are non-negative, discrete.
Big Problem: ω = 0 has infinite multiplicity
Kernel: Electrostatic fields: gradients of all φ ∈ H01 (Ω) (+ harmonic forms).
R
Idea: div E = 0, so we can add a multiple of 0 = Ω div E div F
Regularized formulation: E ∈ X N \ {0} : ∀F ∈ X N :
Z
Z
Z
(RegX )
curl E · curl F + s
div E div F = ω 2 E · F
Ω
Ω
Ω
Energy space: X N = H0 (curl, Ω) ∩ H (div, Ω)
Second order system: curl curl E − s∇ div E = ω 2 E: Strongly elliptic. OK
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
9 / 48
Regularized formulation
Simple variational formulation
E ∈ H0 (curl, Ω)\{0} : ∀F ∈ H0 (curl, Ω) :
Z
curl E · curl F = ω 2
Z
E ·F
Ω
Ω
Galerkin discretization:
Restriction to finite-dimensional subspace Vh , h → 0.
Good: Eigenfrequencies are non-negative, discrete.
Big Problem: ω = 0 has infinite multiplicity
Kernel: Electrostatic fields: gradients of all φ ∈ H01 (Ω) (+ harmonic forms).
R
Idea: div E = 0, so we can add a multiple of 0 = Ω div E div F
Regularized formulation: E ∈ X N \ {0} : ∀F ∈ X N :
Z
Z
Z
(RegX )
curl E · curl F + s
div E div F = ω 2 E · F
Ω
Ω
Ω
Energy space: X N = H0 (curl, Ω) ∩ H (div, Ω)
Second order system: curl curl E − s∇ div E = ω 2 E: Strongly elliptic. OK
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
9 / 48
Regularized formulation
Simple variational formulation
E ∈ H0 (curl, Ω)\{0} : ∀F ∈ H0 (curl, Ω) :
Z
curl E · curl F = ω 2
Z
E ·F
Ω
Ω
Galerkin discretization:
Restriction to finite-dimensional subspace Vh , h → 0.
Good: Eigenfrequencies are non-negative, discrete.
Big Problem: ω = 0 has infinite multiplicity
Kernel: Electrostatic fields: gradients of all φ ∈ H01 (Ω) (+ harmonic forms).
R
Idea: div E = 0, so we can add a multiple of 0 = Ω div E div F
Regularized formulation: E ∈ X N \ {0} : ∀F ∈ X N :
Z
Z
Z
(RegX )
curl E · curl F + s
div E div F = ω 2 E · F
Ω
Ω
Ω
Energy space: X N = H0 (curl, Ω) ∩ H (div, Ω)
Second order system: curl curl E − s∇ div E = ω 2 E: Strongly elliptic. OK
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
9 / 48
Approximation on the square Ω = (0, π) × (0, π), s = 0
Good approximation: Triangular edge elements (15 nodes per side, P1 )
14
12
10
8
6
4
2
0
150
160
170
Eigenvalue
Martin Costabel (Rennes)
180
ωk2
190
200
210
220
vs. rank k
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
10 / 48
Approximation on the square Ω = (0, π) × (0, π), s = 0
Bad approximation: Nodal triangular elements (15 nodes per side, P1 )
14
12
10
8
6
4
2
0
0
20
40
60
80
100
120
140
Eigenvalue ωk2 vs. rank k
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
11 / 48
Regularized formulation in the square
14
12
10
8
6
4
2
0
0
1
2
3
4
5
6
7
ω[s]2 vs. s
Blue circles: computed ω[s]2 with curl-dominant eigenfunctions.
Red stars: computed ω[s]2 with div-dominant eigenfunctions.
div E satisfies s∆ div E = ω 2 div E
Extra eigenvalues: s times Dirichlet eigenvalues.
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
12 / 48
Regularized formulation in the “L”
45
40
35
30
25
20
15
10
5
0
0
1
2
3
4
5
ω[s]2 vs. s
Gray triangles: computed ω[s]2 with indifferent eigenfunctions.
Cyan-Lines: true Maxwell eigenvalues
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
13 / 48
Regularized formulation in the “L”
Error of the first eigenvalue
Valeur propre Maxwell n° 1 pénalisée au bord par λ= 101 ; Maillage 1
1
10
Interp 1
Interp 2
Interp 3
Interp 4
2
0
10
Interp 5
3
4
5
5
6
6
7
7
8
8
9
9
Error
remains
larger than 90%.
10
10
-1
10
1
10
2
10
3
10
Error vs. number of d.o.f.
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
14 / 48
Solution of the source problem, regularized formulation
Exact solution
1
(2nd component E2 = r − 3 cos θ3 )).
curl curl E − ∇ div E = 0
Martin Costabel (Rennes)
Computation with Q3 elements.
in Ω;
E × n = E0
Importance of function spaces for Maxwell equations
on ∂ Ω
Waves 2011, Vancouver 29/07/2011
15 / 48
1991: M. C OSTABEL
A coercive bilinear form for Maxwell’s equations
J. Math. Anal. Appl. 157 (1991) 527–541.
2002: C. H AZARD
Numerical simulation of corner singularities: a paradox in Maxwell-like
problems
Comptes Rendus Mecanique 330 (2002) 57–68.
2002: M. C OSTABEL , M. DAUGE
Weighted regularization of Maxwell equations in polyhedral domains.
A rehabilitation of nodal finite elements
Numer. Math. 93 (2002) 239–277.
2009: A. B UFFA , P. C IARLET J R ., E. J AMELOT
Solving electromagnetic eigenvalue problems in polyhedral domains
with nodal finite elements
Numer. Math. 113 (2009) 497–518.
2011: A. B ONITO, J.-L. G UERMOND
Approximation of the Eigenvalue Problem for Time Harmonic Maxwell
System by Continuous Lagrange Finite Elements
Math. Comp. 80 (2011) 1887–1910.
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
16 / 48
A Tale of Two Function Spaces
An integration by parts formula (Co 1991)
Let Ω ⊂ R3 be a polyhedron. Let u ∈ X N . If u ∈ H 1 (Ω), then
Z
Z
Z
|∇u |2 = | curl u |2 + | div u |2
Ω
Ω
Ω
Corollary
1
2
Define H N = X N ∩ H 1 (Ω). Then H N is a closed subspace of X N .
If Ω is non-convex, then H N 6= X N .
For s > 0, the sesquilinear form
Z
Z
as (E , F ) =
curl E · curl F + s
div E div F
Ω
Ω
is H N -elliptic.
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
17 / 48
A Tale of Two Function Spaces
An integration by parts formula (Co 1991)
Let Ω ⊂ R3 be a polyhedron. Let u ∈ X N . If u ∈ H 1 (Ω), then
Z
Z
Z
|∇u |2 = | curl u |2 + | div u |2
Ω
Ω
Ω
Corollary
1
2
Define H N = X N ∩ H 1 (Ω). Then H N is a closed subspace of X N .
If Ω is non-convex, then H N 6= X N .
For s > 0, the sesquilinear form
Z
Z
as (E , F ) =
curl E · curl F + s
div E div F
Ω
Ω
is H N -elliptic.
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
17 / 48
Two Regularized Variational Formulations
Regularized formulation: E ∈ X N : ∀F ∈ X N :
Z
Z
Z
Z
J ·F
div E div F − ω 2 E · F =
curl E · curl F + s
(RegX )
Ω
Ω
Ω
Ω
Regularized formulation: E ∈ H N : ∀F ∈ H N :
Z
Z
Z
Z
J ·F
div E div F − ω 2 E · F =
curl E · curl F + s
(RegH )
Ω
Consequence
1
2
(For J ∈ L2 (Ω), Ω ⊂ R3 polyhedron)
If ω 2 is not an eigenvalue, then both (RegX ) and (RegH ) have a
unique solution.
Both are solutions of the boundary value problem
curl curl E − s∇ div E − ω 2 E = J
3
Ω
Ω
Ω
in Ω;
E ×n = 0
on ∂ Ω
If Ω is non-convex, then the two solutions are different, in general.
The eigenvalues of (RegX ) and (RegH ) are different, in general.
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
18 / 48
The role of the divergence
In the regularized variational formulation
Z
Z
Z
Z
J ·F
div E div F − ω 2 E · F =
curl E · curl F + s
(RegH )
Ω
Ω
Ω
Ω
one gets an equation for div E by testing with gradients: F = ∇ψ .
Z
Z
Z
J · ∇ψ
div E ∆ψ − ω 2 E · ∇ψ =
s
Ω
Ω
Ω
Z
Z
div E s∆ψ + ω 2 ψ = − div J ψ
⇐⇒
Ω
Ω
For (RegX ) one takes ψ ∈ H01 (∆, Ω).
For (RegH ) one takes ψ ∈ H01 (Ω) ∩ H 2 (Ω).
Lemma
2
Let div J = 0 and ωs not a Dirichlet eigenvalue.
For a solution E of (RegH ) there holds:
If div E ∈ H 1 (Ω), then div E = 0,
and then E is a solution of (RegX ) and hence of the Maxwell problem.
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
19 / 48
Non-density of smooth functions
The space H N is the completion of
{u ∈ C ∞ (Ω) | u × n = 0 on ∂ Ω}
under the norm of H 0 (curl) ∩ H (div)
Z
Z
2
2
ku kX = | curl u | + | div u |2
Ω
Ω
H N 6= X N means that smooth functions are not dense in X N .
Finite element functions, which are piecewise polynomials on a
triangulation of Ω, belong to H N as soon as they belong to X N .
Consequence
Any Maxwell solution or eigenfunction that does not belong to H 1 (Ω)
cannot be approximated by an X N -conforming finite element method that
uses the regularized variational formulation.
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
20 / 48
The Weighted Regularization Method
(Co & Dauge 2002)
Idee : Replace the L2 norm in the regularizing term s
weighted L2 norm.
Ω div E div F
R
by a
w
Weighted Regularized formulation: E ∈ X w
N : ∀F ∈ X N :
Z
Z
Z
Z
2
J ·F
E
·
F
=
w
div
E
div
F
−
ω
curl
E
·
curl
F
+
s
(Regw
)
H
Ω
Ω
Ω
Definition : X w
N = {u ∈ H0 (curl, Ω) |
2
Ω w | div u |
R
We choose
w (x ) = dist(x , S )
Ω
< ∞}
α
where S is the set of singular points (edges, corners) on the boundary.
Lemma
There exists α0 (Ω) < 2 such that
For α0 (Ω) < α < 2 : Smooth functions are dense in X w
N
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
21 / 48
The Weighted Regularization Method
(Co & Dauge 2002)
Theorem
For α0 (Ω) < α < 2 :
If
ω2
s
is not an eigenvalue of the Dirichlet problem for the
weighted Laplacian
div w ∇
then the solutions E of the weighted regularized problem (Regw
H ) satisfy
div E = 0 and are solutions of the Maxwell problem.
Numerical evidence follows...
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
22 / 48
The Weighted Regularization Method
(Co & Dauge 2002)
Theorem
For α0 (Ω) < α < 2 :
If
ω2
s
is not an eigenvalue of the Dirichlet problem for the
weighted Laplacian
div w ∇
then the solutions E of the weighted regularized problem (Regw
H ) satisfy
div E = 0 and are solutions of the Maxwell problem.
Numerical evidence follows...
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
22 / 48
The L shape, WRM
ω[s]2 vs. s
α =0
Degree q = 1
350
300
250
200
150
100
50
0
0
Martin Costabel (Rennes)
10
20
30
40
50
60
70
Importance of function spaces for Maxwell equations
80
90
100
Waves 2011, Vancouver 29/07/2011
23 / 48
The L shape, WRM
ω[s]2 vs. s
α =0
Degree q = 4
350
300
250
200
150
100
50
0
0
Martin Costabel (Rennes)
10
20
30
40
50
60
70
Importance of function spaces for Maxwell equations
80
90
100
Waves 2011, Vancouver 29/07/2011
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The L shape, WRM
ω[s]2 vs. s
α =1
Degree q = 1
350
300
250
200
150
100
50
0
0
Martin Costabel (Rennes)
10
20
30
40
50
60
70
Importance of function spaces for Maxwell equations
80
90
100
Waves 2011, Vancouver 29/07/2011
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The L shape, WRM
ω[s]2 vs. s
α =1
Degree q = 4
350
300
250
200
150
100
50
0
0
Martin Costabel (Rennes)
10
20
30
40
50
60
70
Importance of function spaces for Maxwell equations
80
90
100
Waves 2011, Vancouver 29/07/2011
26 / 48
The L shape, WRM
ω[s]2 vs. s
α =2
Degree q = 1
350
300
250
200
150
100
50
0
0
Martin Costabel (Rennes)
10
20
30
40
50
60
70
Importance of function spaces for Maxwell equations
80
90
100
Waves 2011, Vancouver 29/07/2011
27 / 48
The L shape, WRM
ω[s]2 vs. s
α =2
Degree q = 4
350
300
250
200
150
100
50
0
0
Martin Costabel (Rennes)
10
20
30
40
50
60
70
Importance of function spaces for Maxwell equations
80
90
100
Waves 2011, Vancouver 29/07/2011
28 / 48
The L shape, WRM
Error of the first eigenvalue
Erreurs 1e vp Maxwell du L. Maillage carre unif.
0
10
-1
10
Uniform grids
1
2
3
4
6
8
12
16
2 ss-integ
3 ss-integ
4 ss-integ
6 ss-integ
8 ss-integ
12 ss-integ
16 ss-integ
-2
10
-3
10
2
10
3
10
4
10
Error vs. number of d.o.f.
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
29 / 48
The L shape, WRM
0
10
Error of the first eigenvalue
1
2
2
3
3
-2
10
4
5
4
6
7
8 9 10
Geo. deg. 1
5
-4
10
hp version
Curved L
Geo. deg. 2
6
Geo. deg. 3
9
7
7
Geo. deg. 4
-6
10
Geo. deg. 5
8
10
10
8
Geo. deg. 6
-8
10
2
10
3
10
8
8 4
10
Error vs. number of d.o.f.
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
30 / 48
Recent developments: Mixed formulations
2,w :
2,w :
[Buffa-Jamelot-Ciarlet 2009] E ∈ X w
∀F ∈ X w
N, p ∈ L
N, q ∈ L
Z
curl E · curl F + s
Z
Z
w div E div F − ω 2
Z
Z
w p div F =
ZΩ
Ω
Ω
Ω
E ·F +
J ·F
Ω
w q div E = 0
Ω
[Bonito-Guermont 2011] E ∈ X −α , p ∈ H01 : ∀F ∈ X −α , q ∈ H01 :
Z
curl E · curl F + div E , div F
Ω
−α
− ω2
Z
E ·F +
Ω
Z
Z
∇p · F =
Ω
J ·F
Ω
∇q · E + p, q
Ω
Z
α−1
=0
Here 12 < α < 1.
Discretization of H −α (Ω) scalar product:
ph , qh
Martin Costabel (Rennes)
−α
∼ h2α
Z
Ω
ph qh
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
31 / 48
Recent developments: Mixed formulations
2,w :
2,w :
[Buffa-Jamelot-Ciarlet 2009] E ∈ X w
∀F ∈ X w
N, p ∈ L
N, q ∈ L
Z
curl E · curl F + s
Z
Z
w div E div F − ω 2
Z
Z
w p div F =
ZΩ
Ω
Ω
Ω
E ·F +
J ·F
Ω
w q div E = 0
Ω
[Bonito-Guermont 2011] E ∈ X −α , p ∈ H01 : ∀F ∈ X −α , q ∈ H01 :
Z
curl E · curl F + div E , div F
Ω
−α
− ω2
Z
E ·F +
Ω
Z
Z
∇p · F =
Ω
J ·F
Ω
∇q · E + p, q
Ω
Z
α−1
=0
Here 12 < α < 1.
Discretization of H −α (Ω) scalar product:
ph , qh
Martin Costabel (Rennes)
−α
∼ h2α
Z
Ω
ph qh
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
31 / 48
1
Are function spaces important?
What google has to say
The Dirichlet problem
2
Regularized Variational Formulation of Maxwell Equations
The Maxwell eigenvalue problem
Regularized variational formulations
Numerical observations
Explanation
Weighted Regularization
Numerical evidence for WRM
3
Regularized Boundary Integral Equations for Maxwell Equations
The Electrical Field Integral Equation
The Regularized EFIE
4
Lippmann-Schwinger Equation for the Magnetic Scattering Problem
Electromagnetic Transmission Problems
The Volume Integral Equation
The dielectric problem
The magnetic problem
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
32 / 48
The Electrical Field Integral Equation
Time-harmonic Maxwell transmission problem:
curl E
[E × n]Γ
i ω[H × n]Γ
= iωH ;
curl H
= −i ω E
= 0;
[n · H ]Γ = 0
= j;
[n · E ]Γ = m
+ radiation condition
in R3 \ Γ
on Γ
on Γ
ei ω|x |
:
Representation formula, with Gω (x ) =
4π|x |
Z
Z
E (x ) = Gω (x − y )j (y ) ds(y ) + ∇ Gω (x − y )m(y ) ds(y )
Γ
Γ
Time-harmonic Maxwell scattering problem:
= iωH ;
curl H = −i ω E
in R3 \ Ω
= j0
on Γ = ∂ Ω
+ radiation condition
Continuity condition divΓ j − ω 2 m = 0 & tangential trace on Γ:
(EFIE)
Vω j > + ω12 ∇> Vω divΓ j = j 0
curl E
n × (E × n )
Single layer potential Vω m(x ) =
Martin Costabel (Rennes)
Γ Gω ( x
R
− y )m(y ) ds(y )
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
33 / 48
The Electrical Field Integral Equation
Time-harmonic Maxwell transmission problem:
curl E
[E × n]Γ
i ω[H × n]Γ
= iωH ;
curl H
= −i ω E
= 0;
[n · H ]Γ = 0
= j;
[n · E ]Γ = m
+ radiation condition
in R3 \ Γ
on Γ
on Γ
ei ω|x |
:
Representation formula, with Gω (x ) =
4π|x |
Z
Z
E (x ) = Gω (x − y )j (y ) ds(y ) + ∇ Gω (x − y )m(y ) ds(y )
Γ
Γ
Time-harmonic Maxwell scattering problem:
= iωH ;
curl H = −i ω E
in R3 \ Ω
= j0
on Γ = ∂ Ω
+ radiation condition
Continuity condition divΓ j − ω 2 m = 0 & tangential trace on Γ:
(EFIE)
Vω j > + ω12 ∇> Vω divΓ j = j 0
curl E
n × (E × n )
Single layer potential Vω m(x ) =
Martin Costabel (Rennes)
Γ Gω ( x
R
− y )m(y ) ds(y )
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
33 / 48
The Electrical Field Integral Equation
Time-harmonic Maxwell transmission problem:
curl E
[E × n]Γ
i ω[H × n]Γ
= iωH ;
curl H
= −i ω E
= 0;
[n · H ]Γ = 0
= j;
[n · E ]Γ = m
+ radiation condition
in R3 \ Γ
on Γ
on Γ
ei ω|x |
:
Representation formula, with Gω (x ) =
4π|x |
Z
Z
E (x ) = Gω (x − y )j (y ) ds(y ) + ∇ Gω (x − y )m(y ) ds(y )
Γ
Γ
Time-harmonic Maxwell scattering problem:
= iωH ;
curl H = −i ω E
in R3 \ Ω
= j0
on Γ = ∂ Ω
+ radiation condition
Continuity condition divΓ j − ω 2 m = 0 & tangential trace on Γ:
(EFIE)
Vω j > + ω12 ∇> Vω divΓ j = j 0
curl E
n × (E × n )
Single layer potential Vω m(x ) =
Martin Costabel (Rennes)
Γ Gω ( x
R
− y )m(y ) ds(y )
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
33 / 48
The Electrical Field Integral Equation
Time-harmonic Maxwell transmission problem:
curl E
[E × n]Γ
i ω[H × n]Γ
= iωH ;
curl H
= −i ω E
= 0;
[n · H ]Γ = 0
= j;
[n · E ]Γ = m
+ radiation condition
in R3 \ Γ
on Γ
on Γ
ei ω|x |
:
Representation formula, with Gω (x ) =
4π|x |
Z
Z
E (x ) = Gω (x − y )j (y ) ds(y ) + ∇ Gω (x − y )m(y ) ds(y )
Γ
Γ
Time-harmonic Maxwell scattering problem:
= iωH ;
curl H = −i ω E
in R3 \ Ω
= j0
on Γ = ∂ Ω
+ radiation condition
Continuity condition divΓ j − ω 2 m = 0 & tangential trace on Γ:
(EFIE)
Vω j > + ω12 ∇> Vω divΓ j = j 0
curl E
n × (E × n )
Single layer potential Vω m(x ) =
Martin Costabel (Rennes)
Γ Gω ( x
R
− y )m(y ) ds(y )
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
33 / 48
The EFIE: Principal properties
The EFIE on Γ
(EFIE)
Vω j
>
+
1
∇> Vω divΓ j = j 0
ω2
Lemma (Nedelec, second half of 20th century)
If ω 2 is not a Dirichlet eigenvalue in Ω, then
Cω : j 7→ Vω j
>
+
1
ω2
∇> Vω divΓ j
1
is an isomorphism between H − 2 (divΓ , Γ) and its dual space. One has
n × Cω
2
1
4
= I − Mω2
1
where Mω is a compact operator in H − 2 (divΓ , Γ) if Γ is smooth.
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
34 / 48
The EFIE: Problems
Good: (EFIE) is given by a non-degenerate quadratic form
Big problem : The principal part of Cω is indefinite.
No strong ellipticity in the sense of pseudodifferential operators, no
convergence of arbitrary Galerkin methods.
Two ways out, as before:
1
2
Construct special finite elements, and prove a generalized strong
ellipticity property, or
Regularize
We describe the regularization method introduced by MacCamy–Stephan
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
35 / 48
The EFIE: Problems
Good: (EFIE) is given by a non-degenerate quadratic form
Big problem : The principal part of Cω is indefinite.
No strong ellipticity in the sense of pseudodifferential operators, no
convergence of arbitrary Galerkin methods.
Two ways out, as before:
1
2
Construct special finite elements, and prove a generalized strong
ellipticity property, or
Regularize
We describe the regularization method introduced by MacCamy–Stephan
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
35 / 48
The EFIE: Problems
Good: (EFIE) is given by a non-degenerate quadratic form
Big problem : The principal part of Cω is indefinite.
No strong ellipticity in the sense of pseudodifferential operators, no
convergence of arbitrary Galerkin methods.
Two ways out, as before:
1
2
Construct special finite elements, and prove a generalized strong
ellipticity property, or
Regularize
We describe the regularization method introduced by MacCamy–Stephan
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
35 / 48
The regularized EFIE (MacCamy & Stephan 1982)
Write the EFIE as a system
Vω j
>
+ ∇> Vω m = j 0
Vω divΓ j − ω 2 Vω m = 0
This is still essentially indefinite.
Now multiply the first equation by divΓ and subtract:
Vω j
(EFIEreg )
>
+ ∇> Vω m = j 0
Kω j − (∆Γ + ω 2 )Vω m = − divΓ j 0
Here ∆Γ is the Laplace-Beltrami operator, and
Kω = Vω divΓ − divΓ Vω is an operator of order −1 if Γ is smooth.
Theorem (MacCamy–Stephan)
The system (EFIEreg ) is a strongly elliptic system of pseudodifferential
operators. It defines a Fredholm operator of index 0
s− 21
H>
1
s+ 1
1
(Γ) × H s+ 2 (Γ) → H > 2 (Γ) × H s− 2 (Γ)
Any Galerkin scheme for its approximation is stable in
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
∀s ∈ R
1
−1
H > 2 (Γ) × H 2 (Γ).
Waves 2011, Vancouver 29/07/2011
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The regularized EFIE (MacCamy & Stephan 1982)
Write the EFIE as a system
Vω j
>
+ ∇> Vω m = j 0
Vω divΓ j − ω 2 Vω m = 0
This is still essentially indefinite.
Now multiply the first equation by divΓ and subtract:
Vω j
(EFIEreg )
>
+ ∇> Vω m = j 0
Kω j − (∆Γ + ω 2 )Vω m = − divΓ j 0
Here ∆Γ is the Laplace-Beltrami operator, and
Kω = Vω divΓ − divΓ Vω is an operator of order −1 if Γ is smooth.
Theorem (MacCamy–Stephan)
The system (EFIEreg ) is a strongly elliptic system of pseudodifferential
operators. It defines a Fredholm operator of index 0
s− 21
H>
1
s+ 1
1
(Γ) × H s+ 2 (Γ) → H > 2 (Γ) × H s− 2 (Γ)
Any Galerkin scheme for its approximation is stable in
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
∀s ∈ R
1
−1
H > 2 (Γ) × H 2 (Γ).
Waves 2011, Vancouver 29/07/2011
36 / 48
References
1983: R. M AC C AMY, E. S TEPHAN
A boundary element method for an exterior problem for
three-dimensional Maxwell?s equations
Applicable Anal. 16 (1983) 141–163.
1984: R. M AC C AMY, E. S TEPHAN
Solution procedures for three-dimensional eddy current problems
J. Math. Anal. Appl. 101 (1984) 348–379.
1998: N. H EUER
Preconditioners for the boundary element method for solving the
electric screen problem
Pitman Res. Notes Math. Ser. 379 (1998) 106–110.
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
37 / 48
Another Tale of Two Function Spaces
We consider scattering by an open surface Γ (Screen problem)
1
1
Given j 0 ∈ H − 2 (curl> , Γ), then for any ω > 0 the Maxwell scattering
problem and (EFIE) each have a unique solution
E ∈ H loc (curl, R3 \ Γ) + radiation condition
e
j ∈H
− 12
(divΓ , Γ)
1
2
If j 0 ∈ H >2 (Γ), then also (EFIEreg ) has a unique solution
e
(j , m ) ∈ H
3
− 12
1
e 2 (Γ)
(Γ) × H
The solution of (EFIEreg ) can be approximated by any conforming
finite element method [Stephan 1984, Heuer 1996].
Trap : The solutions j of (EFIE) and of (EFIEreg ) are different, in general,
1
e 2 (Γ).
and the Maxwell solution does not satisfy divΓ j = ω 2 m ∈ H
1
Edge singularity: m ∼ r − 2 6∈ L2 (Γ)
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
38 / 48
Another Tale of Two Function Spaces
We consider scattering by an open surface Γ (Screen problem)
1
1
Given j 0 ∈ H − 2 (curl> , Γ), then for any ω > 0 the Maxwell scattering
problem and (EFIE) each have a unique solution
E ∈ H loc (curl, R3 \ Γ) + radiation condition
e
j ∈H
− 12
(divΓ , Γ)
1
2
If j 0 ∈ H >2 (Γ), then also (EFIEreg ) has a unique solution
e
(j , m ) ∈ H
3
− 12
1
e 2 (Γ)
(Γ) × H
The solution of (EFIEreg ) can be approximated by any conforming
finite element method [Stephan 1984, Heuer 1996].
Trap : The solutions j of (EFIE) and of (EFIEreg ) are different, in general,
1
e 2 (Γ).
and the Maxwell solution does not satisfy divΓ j = ω 2 m ∈ H
1
Edge singularity: m ∼ r − 2 6∈ L2 (Γ)
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
38 / 48
1
Are function spaces important?
What google has to say
The Dirichlet problem
2
Regularized Variational Formulation of Maxwell Equations
The Maxwell eigenvalue problem
Regularized variational formulations
Numerical observations
Explanation
Weighted Regularization
Numerical evidence for WRM
3
Regularized Boundary Integral Equations for Maxwell Equations
The Electrical Field Integral Equation
The Regularized EFIE
4
Lippmann-Schwinger Equation for the Magnetic Scattering Problem
Electromagnetic Transmission Problems
The Volume Integral Equation
The dielectric problem
The magnetic problem
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
39 / 48
References
2010: M. C OSTABEL , E. DARRIGRAND, E. KONÉ
Volume and surface integral equations for electromagnetic scattering
by a dielectric body
J. Comput. Appl. Math. 234 (2010) 1817–1825.
2007: A. K IRSCH
An integral equation approach and the interior transmission problem
for Maxwell’s equations
Inverse Probl. Imaging 1 (2007) 159–179.
2010: A. K IRSCH , A. L ECHLEITER
The operator equations of Lippmann-Schwinger type for acoustic and
electromagnetic scattering problems in L2
Appl. Anal. 88 (2010) 807–830.
2011: M. C OSTABEL , E. DARRIGRAND, H. S AKLY
On the essential spectrum of the volume integral operator in
electromagnetic scattering
In preparation.
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
40 / 48
Electromagnetic scattering by a penetrable homogeneous object
µ = µr in Ω, µ = 1 in R3 \ Ω, ε = εr in Ω, ε = 1 in R3 \ Ω.
Maxwell equations
curl E = i ω µ H ;
curl H = −i ωε E + J
hold in R3 in the distributional sense (+ radiation condition).
supp J compact in R3 \ Ω.
=⇒ Transmission conditions on Γ = ∂ Ω:
[E × n]Γ
[H × n]Γ
= 0;
= 0;
[n · µ H ]Γ
[n · ε E ]Γ
= 0
= 0
Lippmann-Schwinger equation: One considers the obstacle as a
perturbation: curl µ1 curl E − ω 2 ε E = i ω J ⇔
curl curl E − ω 2 E = i ω J − ω 2 pE + curl q curl E
with p = (1 − εr )χΩ , q = (1 − µ1r )χΩ .
The right-hand side has compact support: Convolution with fundamental
solution of curl curl −ω 2 :
gω =
Martin Costabel (Rennes)
1
ω2
∇ div +1 Gω ;
Gω ( x ) =
Importance of function spaces for Maxwell equations
ei ω|x |
4π|x |
Waves 2011, Vancouver 29/07/2011
41 / 48
Electromagnetic scattering by a penetrable homogeneous object
µ = µr in Ω, µ = 1 in R3 \ Ω, ε = εr in Ω, ε = 1 in R3 \ Ω.
Maxwell equations
curl E = i ω µ H ;
curl H = −i ωε E + J
hold in R3 in the distributional sense (+ radiation condition).
supp J compact in R3 \ Ω.
=⇒ Transmission conditions on Γ = ∂ Ω:
[E × n]Γ
[H × n]Γ
= 0;
= 0;
[n · µ H ]Γ
[n · ε E ]Γ
= 0
= 0
Lippmann-Schwinger equation: One considers the obstacle as a
perturbation: curl µ1 curl E − ω 2 ε E = i ω J ⇔
curl curl E − ω 2 E = i ω J − ω 2 pE + curl q curl E
with p = (1 − εr )χΩ , q = (1 − µ1r )χΩ .
The right-hand side has compact support: Convolution with fundamental
solution of curl curl −ω 2 :
gω =
Martin Costabel (Rennes)
1
ω2
∇ div +1 Gω ;
Gω ( x ) =
Importance of function spaces for Maxwell equations
ei ω|x |
4π|x |
Waves 2011, Vancouver 29/07/2011
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Lippmann-Schwinger equation
Representation of E in R3 by volume integrals over Ω
E = ω 2 gω ∗ (pE ) + gω ∗ (curl q curl E ) + E inc
=⇒ Volume integral equation in Ω
E = pAω E + qBω E + E inc
with
Z
Z
Aω E (x ) = −∇ div Gω (x − y )E (y ) dy − ω 2 Gω (x − y )E (y ) dy
Ω
Z Ω
Bω E (x ) = curl Gω (x − y ) curl E (y ) dy
Ω
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
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The dielectric scattering problem: q = 0
Volume integral equation:
Results
E − pAω E = E inc
(Co & E. Darrigrand & E.H. Koné 2009)
1
The operator Aω can be extended to L2 (Ω) as a bounded operator.
2
It has H (curl, Ω) and H (div, Ω) as invariant subspaces.
3
4
5
For E inc in H (curl, Ω) ∩ H (div, Ω), the integral equation in L2 has the
same solutions as in H (curl, Ω) or in H (div, Ω).
Aω − A0 is compact in L2 (Ω).
Sp(A0 ) ⊂ [0, 1]
0 and 1 are eigenvalues of infinite multiplicity of A0
1
is accumulation point of eigenvalues, and
2
Spess (A0 ) = {0, 12 , 1} if Γ is smooth
The dielectric scattering problem can be solved by solving the volume
integral equation in L2 (Ω).
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
43 / 48
The magnetic scattering problem: p = 0 (Co & E. Darrigrand & H. Sakly)
Volume integral equation:
The integral operator
E − qBω E = E inc
Bω E (x ) = curl
Z
Ω
Gω (x − y ) curl E (y ) dy
is bounded from H (curl, Ω) to itself and to H (div 0, Ω).
For E ∈ C ∞
0 (Ω) one has
Bω E = curl Gω ∗ curl E = curl curl Gω ∗ E
= ∇ div Gω ∗ E − (∆ + ω 2 )Gω ∗ E + ω 2 Gω ∗ E
= E − Aω E
Proposition
(Kirsch & Lechleiter 2010)
The operator Bω can be extended to L2 (Ω) as a bounded operator.
Trap alert !
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
44 / 48
The magnetic scattering problem: p = 0 (Co & E. Darrigrand & H. Sakly)
Volume integral equation:
The integral operator
E − qBω E = E inc
Bω E (x ) = curl
Z
Ω
Gω (x − y ) curl E (y ) dy
is bounded from H (curl, Ω) to itself and to H (div 0, Ω).
For E ∈ C ∞
0 (Ω) one has
Bω E = curl Gω ∗ curl E = curl curl Gω ∗ E
= ∇ div Gω ∗ E − (∆ + ω 2 )Gω ∗ E + ω 2 Gω ∗ E
= E − Aω E
Proposition
(Kirsch & Lechleiter 2010)
The operator Bω can be extended to L2 (Ω) as a bounded operator.
Trap alert !
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
44 / 48
The magnetic scattering problem: p = 0 (Co & E. Darrigrand & H. Sakly)
Volume integral equation:
The integral operator
E − qBω E = E inc
Bω E (x ) = curl
Z
Ω
Gω (x − y ) curl E (y ) dy
is bounded from H (curl, Ω) to itself and to H (div 0, Ω).
For E ∈ C ∞
0 (Ω) one has
Bω E = curl Gω ∗ curl E = curl curl Gω ∗ E
= ∇ div Gω ∗ E − (∆ + ω 2 )Gω ∗ E + ω 2 Gω ∗ E
= E − Aω E
Proposition
(Kirsch & Lechleiter 2010)
The operator Bω can be extended to L2 (Ω) as a bounded operator.
Trap alert !
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
44 / 48
A Third Tale of Two Function Spaces
Theorem 1
Solving the volume integral equation
E − qBω E = E inc
in H (curl, Ω) is equivalent to the magnetic Maxwell scattering problem.
Theorem 2
bω : L2 (Ω) → L2 (Ω) be the extended operator. Solving the volume
Let B
integral equation
bω E = E inc
E − qB
in L2 (Ω) gives the Maxwell equations in R3 \ Γ with the transmission
conditions
[n · H ]Γ = 0
[ µ1 E × n]Γ = 0 ;
[H × n]Γ = 0 ;
[n · E ]Γ = 0
Wrong !
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
45 / 48
A Third Tale of Two Function Spaces
Theorem 1
Solving the volume integral equation
E − qBω E = E inc
in H (curl, Ω) is equivalent to the magnetic Maxwell scattering problem.
Theorem 2
bω : L2 (Ω) → L2 (Ω) be the extended operator. Solving the volume
Let B
integral equation
bω E = E inc
E − qB
in L2 (Ω) gives the Maxwell equations in R3 \ Γ with the transmission
conditions
[n · H ]Γ = 0
[ µ1 E × n]Γ = 0 ;
[H × n]Γ = 0 ;
[n · E ]Γ = 0
Wrong !
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
45 / 48
A Third Tale of Two Function Spaces
Theorem 1
Solving the volume integral equation
E − qBω E = E inc
in H (curl, Ω) is equivalent to the magnetic Maxwell scattering problem.
Theorem 2
bω : L2 (Ω) → L2 (Ω) be the extended operator. Solving the volume
Let B
integral equation
bω E = E inc
E − qB
in L2 (Ω) gives the Maxwell equations in R3 \ Γ with the transmission
conditions
[n · H ]Γ = 0
[ µ1 E × n]Γ = 0 ;
[H × n]Γ = 0 ;
[n · E ]Γ = 0
Wrong !
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
45 / 48
Explanation : For E ∈ H (curl, Ω), one has
Z
bω E = Bω E + curl Gω (x − y )E (y ) × n(y ) ds(y )
B
Γ
The latter term does not have a continuous extension to L2 (Ω).
Proposition 1
The operator Bω cannot be extended from H (curl, Ω) to L2 (Ω) as a
bounded operator.
Proposition 2
Although on C ∞
0 (Ω) we have
Bω = I − Aω ,
the commutator of Aω and Bω on H (curl, Ω) is not compact.
=⇒ No joint essential spectrum of Aω and Bω in the Lippmann-Schwinger
operator
I − pAω − qBω
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
46 / 48
Explanation : For E ∈ H (curl, Ω), one has
Z
bω E = Bω E + curl Gω (x − y )E (y ) × n(y ) ds(y )
B
Γ
The latter term does not have a continuous extension to L2 (Ω).
Proposition 1
The operator Bω cannot be extended from H (curl, Ω) to L2 (Ω) as a
bounded operator.
Proposition 2
Although on C ∞
0 (Ω) we have
Bω = I − Aω ,
the commutator of Aω and Bω on H (curl, Ω) is not compact.
=⇒ No joint essential spectrum of Aω and Bω in the Lippmann-Schwinger
operator
I − pAω − qBω
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
46 / 48
Conclusion and Perspectives
Advice :
When you find a brilliant new algorithm for Maxwell’s equations, be sure to
check your function spaces !
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
47 / 48
Conclusion and Perspectives
Advice :
When you find a brilliant new algorithm for Maxwell’s equations, be sure to
check your function spaces !
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
47 / 48
Thank you for your attention!
Martin Costabel (Rennes)
Importance of function spaces for Maxwell equations
Waves 2011, Vancouver 29/07/2011
48 / 48