Encounters with Maxwell Equations Martin Costabel IRMAR, Universit - - PowerPoint PPT Presentation

Encounters with Maxwell Equations

Martin Costabel IRMAR, Université de Rennes 1 Analysis and Numerics of Acoustic and Electromagnetic Problems Linz, 17–22 October 2016

Martin Costabel (Rennes) Encounters with Maxwell Equations Linz, 17/10/2016 1 / 38

Stories of Encounters with Maxwell equations

Four stories were planned:

1

Strong Ellipticity: Beginning 1982

2

Maxwell Corner Singularities: Beginning 1997

3

Approximation of Eigenvalue Problems: Beginning 2002

4

Volume Integral Equations: Beginning 2005 Today: Only Stories

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Story 1: Strong ellipticity

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The Beginning

• R. C. MacCamy and E. Stephan.

A boundary element method for an exterior problem for three-dimensional Maxwell’s equations. Applicable Anal., 16(2):141–163, 1983.

• R. C. MacCamy and E. Stephan.

Solution procedures for three-dimensional eddy current problems.

• J. Math. Anal. Appl., 101(2):348–379, 1984.
• M. Costabel and E. P

. Stephan. Strongly elliptic boundary integral equations for electromagnetic transmission problems.

• Proc. Roy. Soc. Edinburgh Sect. A, 109(3-4):271–296, 1988.
• M. Costabel.

A coercive bilinear form for Maxwell’s equations.

• J. Math. Anal. Appl., 157(2):527–541, 1991.

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Oberwolfach 1988 Richard C. MacCamy

26/09/1925 – 6/07/2011

Suri, Hsiao, Stephan, Costabel, Wendland, MacCamy

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Context: Numerical approximation of pseudodifferential equations

Galerkin approximation (Var) u ∈ X : a(u, v) = f, v ∀ v ∈ X (Varh) uh ∈ Xh : a(uh, v) = f, v ∀ v ∈ Xh Crucial Property: Strong ellipticity Strong ellipticity = ⇒ Gårding inequality, “a pos. def. + compact” Result: Every Galerkin method is stable and convergent. Class of interest: Pseudodifferential operators Pseudodifferential operator A of order α with positive principal symbol = ⇒ Strong ellipticity in X = Hα/2(Ω), a(u, v) = Au, v. Activity: Find relevant examples. Found: Acoustics, elasticity, fluid dynamics,... What about electromagnetics? Need to generalize strong ellipticity.

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BIE for time-harmonic Maxwell equations

The EFIE V τj +

1 κ2 ∇τV divτ j = E0

Vψ(x) =

• ∂Ω

eiκ|x−y| 4π|x−y|ψ(y)ds(y) , κ > 0

Bilinear form (u, v tangential vector fields) a(u, v) = Vu, v −

1 κ2 V divτ u, divτ v

• V is strongly elliptic (of order α = −1), but a is indefinite!
• Principal part (order α = +1) not elliptic!

Idea (“T-coercivity”): Replace a(u, v) by a(u, Tv) with some T : X → X and prove that a(·, T·) is pos. def + compact.

Two concretizations

1

Keep a(·, ·) in the computations, use T only in the theory. This works if one can find Th : Xh → Xh that is “close” to T. Works for EFIE if T is defined by Hodge decomposition and Xh are specially crafted finite elements (edge elements, discrete differential forms). Also: X = H−1/2(divτ, ∂Ω).(∗)

2

Change the equation: Replace A by T ∗A. Then any Galerkin method will still do.

(∗) This would be another story, in which I only participated at the beginning.

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The beginning of another story, later

• A. Buffa, M. Costabel, and D. Sheen.

On traces for H(curl, Ω) in Lipschitz domains.

• J. Math. Anal. Appl., 276(2):845–867, 2002.
• A. Buffa, M. Costabel, and C. Schwab.

Boundary element methods for Maxwell’s equations on non-smooth domains.

• Numer. Math., 92(4):679–710, 2002.
• M. Costabel and C. Safa.

A boundary integral formulation of antenna problems suitable for nodal-based wavelet approximations. In Numerical mathematics and advanced applications, pages 265–271. Springer Italia, Milan, 2003.

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Back to the MacCamy-Stephan boundary integral equation

In EFIE, introduce a scalar potential m as additional unknown, and add a second equation stating that div E = 0 on the boundary. Use (∆ + κ2)Vm = 0 in Ω.

• V τ j + ∇τ Vm = E0

V divτ j − κ2Vm = 0 Problem: This is an elliptic (in the sense of ADN) system of pseudodifferential operators on H1/2 × H−1/2, but it is not strongly elliptic. Principal symbol

1 |ξ|

  1 iξ1 1 iξ2 −iξ1 −iξ2   Idea: Modify the system by “Gauss elimination”: Subtract divτ times the first equation from the second. If ∂Ω is smooth, the commutator J = [divτ, V τ] is of lower order −1. The system becomes

• V τ j + ∇τ Vm = E0

Jj + (∆τ + κ2)Vm = − divτ E0 Principal symbol

1 |ξ|

  1 iξ1 1 iξ2 |ξ|2   triangular and pos. def. of orders (−1, −1, +1). This works (for smooth boundaries): Any Galerkin method is stable. No special FEM needed.

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Interpretation, generalization to transmission problem

Trying to understand: The “Dirichlet data” (Eτ, div E) were changed into (Eτ, div E − divτ Eτ). This mapping TDir defines an isomorphism of H−1/2(div) × H−1/2 to itself and also of H1/2

τ

× H−1/2 to itself. Strong ellipticity was obtained in the latter space. There is a corresponding mapping for the “Neumann” data TNeu : (En, n × curl E) → (En, n × curl E + ∇τ En) Consider now the Maxwell transmission problem in regularized form: (∆ + k2)u = 0 in Ω ∪ Ωc [uτ] = u0

τ , [εun] = ε2u0 n , [ 1 µ n × curl u] = 1 µ2 n × u0 , [λ div u] = λ2 div u0

• n ∂Ω

Here κ, ε, µ, λ are piecewise constant, taking values κ1 in Ω and κ2 in Ωc. The constants κ, ε, µ are physical, λ is suitably chosen so as to enforce div u = 0 if the incident field u0 is divergence free. We introduce Cauchy data corresponding to these jump conditions (w, φ, ψ, v) = (uτ, λ div u, − 1

µ n × curl u, εun)

and finally the modification operator TCauchy : (w, φ, ψ, v) → (w, ηφ − divτ w, ψ + θ∇τ v, θv) where η, θ ∈ C are yet to be chosen.

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The regularized Maxwell transmission problem

Theorem [Costabel-Stephan 1988] Consider the “direct” boundary integral equation method for the Maxwell transmission problem, where the system of boundary integral operators A is the difference between the exterior and interior Calderón projectors for the vector Helmholtz equations, defined using the modified Cauchy data as above. If Ω is smooth, then under rather general natural conditions on κ, ε, µ, the parameters λ, η, θ can be chosen such that A is a strongly elliptic system of pseudodifferential operators defined on the boundary function space H

1/2 τ

× H−1/2 × H−1/2

τ

× H1/2. Proof by explicit calculation of the principal symbol.

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Replacing calculations of symbols by Green’s formula

Lemma [Co 1991] Let Ω ⊂ R3 be a bounded smooth domain and u, v ∈ C2(Ω). Then

• Ω
• curl u · curl v + div u div v
• + c(u, v) =
• Ω

∇u · ∇v + b(u, v) where c(u, v) =

• ∂Ω
• ∇τun · vτ − divτ uτvn
• b(u, v) =
• ∂Ω
• (uτ · ∇n) · vτ) + div n un vn
• On the left hand side, we recognize the modification of the Cauchy data:

n × curl u → n × curl u + ∇τun and div u → div u − divτ uτ On the right hand side, we find the curvature of the boundary, which for a C2 boundary consists of bounded functions. Corollary

For a bounded C2 domain Ω, the bilinear form on the left hand side is strongly elliptic on H1(Ω). This allows to prove the strong ellipticity of the boundary integral system studied in [Costabel-Stephan 1988] by integration by parts, without computing symbols of pseudodifferential operators.

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What about non-smooth domains?

In the paper [Co 1991], there was a remark in the last paragraph that for a polyhedron and functions that are in X = H(curl) ∩ H(div) and have either vanishing tangential component (XN) or vanishing normal component (XT ) on the boundary, this integration-by-parts formula simplifies considerably, if one knows that the function is in H1(Ω). Corollary 2 [Co 1991] If Ω is a polyhedron and u ∈ HN ∪ HT , then

• Ω
• | curl u|2 + | div u|2

=

• Ω

|∇u|2 As a consequence, HN = XN ∩ H1 and HT = XT ∩ H1 are closed subspaces of XN and XT , respectively, of infinite codimension if Ω is a non-convex polyhedron in R3. This remark turned out to be the only part of that paper that wasn’t forgotten... It implies that approximation of elements of XN \ HN or XT \ HT by conforming finite elements is impossible. And this had consequences...

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First consequence: Study Maxwell singularities, in particular non H1

• M. Costabel.

A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains.

• Math. Methods Appl. Sci., 12(4):365–368, 1990.
• M. Costabel and M. Dauge.

Singularities of electromagnetic fields in polyhedral domains.

• Arch. Ration. Mech. Anal., 151(3):221–276, 2000.
• M. Costabel, M. Dauge, and S. Nicaise.

Singularities of Maxwell interface problems. M2AN Math. Model. Numer. Anal., 33(3):627–649, 1999.

• M. Costabel, M. Dauge, and S. Nicaise.

Singularities of eddy current problems. M2AN Math. Model. Numer. Anal., 37(5):807–831, 2003. But this is another story...

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Second consequence: Find out if this is important for numerics

• M. Costabel and M. Dauge.

Maxwell and Lamé eigenvalues on polyhedra.

• Math. Methods Appl. Sci., 22(3):243–258, 1999.
• M. Costabel and M. Dauge.

Computation of resonance frequencies for Maxwell equations in non-smooth domains. In Topics in computational wave propagation, volume 31 of Lect. Notes Comput. Sci. Eng., pages 125–161. Springer, Berlin, 2003.

It turns out: Yes. If you regularize time-harmonic Maxwell equations in order to make them strongly elliptic and use standard finite elements for discretization, you can get not only bad approximations, but — worse — good approximations of wrong results. Regularized formulation: E ∈ X N \ {0} : ∀F ∈ X N :

• Ω

curl E · curl F + s

• Ω

div E div F = ω2

• Ω

E · F Energy space: X N = H0(curl, Ω) ∩ H(div, Ω) Second order system: curl curl E − s∇ div E = ω2E: Strongly elliptic. OK Big problem: Finite elements perform automatically the variational formulation in HN = XN ∩ H1. − → Different spectrum.

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Expected behavior: Regularized formulation in the square

1 2 3 4 5 6 7 2 4 6 8 10 12 14

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 = ω2 div E Extra eigenvalues: s times Dirichlet eigenvalues.

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Observed behavior: Regularized formulation in the “L ”

1 2 3 4 5 5 10 15 20 25 30 35 40 45

ω[s]2 vs. s Gray triangles: computed ω[s]2 with indifferent eigenfunctions. Cyan-Lines: true Maxwell eigenvalues These eigenvalues belong to Lamé, not to Maxwell.

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Third consequence: Find a solution for the problem of non-convergence

• M. Costabel and M. Dauge.

Weighted regularization of Maxwell equations in polyhedral domains. A rehabilitation

• f nodal finite elements.
• Numer. Math., 93(2):239–277, 2002.
• M. Costabel, M. Dauge, and C. Schwab.

Exponential convergence of hp-FEM for Maxwell equations with weighted regularization in polygonal domains.

• Math. Models Methods Appl. Sci., 15(4):575–622, 2005.

Idea: The problem comes from the fact that the regularized Maxwell bilinear form a(u, v) =

• Ω
• curl u · curl v + s div u div v
• defines a space XN = H0(curl) ∩ H(div) in which smooth functions (and piecewise

polynomials) are not dense. The term

• Ω div u div v was rather arbitrary, because div u = 0 for Maxwell solutions. It can

be replaced by a different bilinear form, that is, the norm div uL2(Ω) can be replaced by a norm div uY with a space Y satisfying the two requirements:

1

The corresponding space X Y

N = H0(curl) ∩ {u ∈ L2 | div u ∈ Y}

is still compactly embedded in L2(Ω).

2

C∞(Ω) ∩ X Y

N is dense in X Y N .

Martin Costabel (Rennes) Encounters with Maxwell Equations Linz, 17/10/2016 18 / 38

Weighted regularization

We proposed to choose Y as a weighted L2 space: q2

Y =

• Ω

ρ2γ|q|2 , ρ distance to the non-convex edges and corners Theorem [Costabel-Dauge 2002] Let Ω be a polygon in R2 or a polyhedron in R3. Define DY(∆) = {q ∈ H1

0(Ω) | ∆q ∈ Y}. Then

(i) Decomposition X Y

N = HN + ∇DY(∆).

(ii) If H2 ∩ H1

0(Ω) is dense in DY(∆), then condition

2 is satisfied.

(iii) There exists 0 < γ∗ < 1 such that for weight exponents γ∗ < γ < 1, conditions

1 and 2 are satisfied.

γ∗ = max{1 −

π ωe , 1 2 − λDir c

} .

As a consequence, for this choice of weights, any conforming finite element Galerkin method for the regularized Maxwell system (source problem or eigenvalue problem) converges in X Y

N .

More recently, results with other choices of Y have appeared: e.g. Y = H−s for 0 < s < 1, or a discretized version thereof, · Y,h = hs · L2 [Bonito-Guermond MathComp 2011]

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Weighted Regularization in the “L ”

Computations with Q10 elements on refined mesh. Legend: Blue circles: computed ω[s]2 with curl-dominant eigenfunctions. Red stars: computed ω[s]2 with div-dominant eigenfunctions. Gray triangles: computed ω[s]2 with indifferent eigenfunctions. Cyan Lines: true Maxwell eigenvalues (calculated from scalar Neumann problem)

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The ideal: Computations in the square

1 2 3 4 5 6 7 2 4 6 8 10 12 14

ω[s]2 vs. s

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The unweighted reality: Computations in the L (γ = 0)

1 2 3 4 5 5 10 15 20 25 30 35 40 45

ω[s]2 vs. s

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Towards the ideal: WRM in the L (γ = 0.35)

2 4 6 8 10 5 10 15 20 25 30 35 40 45

ω[s]2 vs. s

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Towards the ideal: WRM in the L (γ = 0.5)

2 4 6 8 10 5 10 15 20 25 30 35 40 45

ω[s]2 vs. s

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The ideal recovered: WRM in the L (γ = 1)

5 10 15 20 5 10 15 20 25 30 35 40 45

ω[s]2 vs. s

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Story 3: Approximation of Eigenvalue Problems

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References

• M. Costabel and M. Dauge.

Computation of resonance frequencies for Maxwell equations in non-smooth domains. In Topics in computational wave propagation, volume 31 of Lect. Notes

• Comput. Sci. Eng., pages 125–161. Springer, Berlin, 2003.
• D. Boffi, L. Demkowicz, and M. Costabel.

Discrete compactness for p and hp 2D edge finite elements.

• Math. Models Methods Appl. Sci., 13(11):1673–1687, 2003.
• D. Boffi, M. Costabel, M. Dauge, and L. Demkowicz.

Discrete compactness for the hp version of rectangular edge finite elements. SIAM J. Numer. Anal., 44(3):979–1004, 2006.

• D. Boffi, M. Costabel, M. Dauge, L. Demkowicz, and R. Hiptmair.

Discrete compactness for the p-version of discrete differential forms. SIAM J. Numer. Anal., 49(1):135–158, 2011.

• M. Costabel and A. McIntosh.

On Bogovski˘ ı and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains.

• Math. Z., 265(2):297–320, 2010.

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Oberwolfach 2004 Alan G. R. McIntosh

17/01/1942 – 8/08/2016

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Motivation

Observation in our computations for problems with singularities: Higher order polynomials give very efficient approximations,

• ne should even get exponential convergence for the hp version of the finite

element method. Open problem (until 2008): Spectrally correct approximation of Maxwell eigenvalues in the p or hp version.

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Approximation on the square Ω = (0, π) × (0, π)

Another bad approximation: One square element ((Q8)2 ∩ H0(curl, Ω))

10 20 30 40 50 60 70 2 4 6 8 10 12 14

Wrong multiplicities ! Too many discrete divergence-free functions

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Approximation on the square Ω = (0, π) × (0, π)

Spurious eigenfunctions in (Q8)2 ∩ H0(curl, Ω): Graph of x-component

1 2 3 4 1 2 3 4 0.2 0.4 0.6 0.8 1 1 2 3 4 1 2 3 4 0.4 0.2 0.2 0.4 0.6 0.8 1

True first eigenfunction Spurious eigenfunction

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What was missing for the proof? Since [Kikuchi 1989] and [Caorsi-Fernandes-Raffetto 2000] one knows that a decisive property is a discrete compactness property or something equivalent. This property had been proved for the h version FEM, but not for the p version. In [Boffi-Demkowicz-Costabel 2003], we first tried hard to prove a certain uniform inf-sup condition for the curl operator that would have implied the discrete

• compactness. It turned out that this was impossible to prove.

Numerical tests showed that the inf-sup constant tends to zero as O(p−1/2), and for an analogous 1D problem we could find the inf-sup constant explicitly, with the behavior of O(p−1/2). We proved that this was sufficient to get a spectrally correct approximation. In [Bo-Co-Da-De 2006], we could prove this behavior for special situations in 2D. We attacked the 3D problem in 2008, while Bo-Co-Da were visiting De in Austin. It turned out that the proof of one little lemma would make the whole proof go

• through. Ralf Hiptmair had arrived at this point, too.

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A Crucial Lemma

Lemma Situation: K reference element, p ∈ N, W p(K) edge elements of degree p, 0 < r < 1. If z ∈ Hr(K) such that curl z ∈ curl W p(K), then ∃ v ∈ W p(K) : curl z = curl v

• r also

z = v + ∇φ, such that there is a constant C independent of p with v r + φ 1+r ≤ C z r Proof: Use regularized Poincaré operators z = Rcurl curl z + ∇R∇z = v + ∇φ Continuity: Rcurl : Hr−1(K) → Hr(K) , R∇ : Hr(K) → Hr+1(K) These operators were analyzed in [Costabel-McIntosh 2010]. The complete proof of spectrally correct approximation of the p version of edge elements was published in [Boffi-Costabel-Dauge-Demkowicz-Hiptmair 2011].

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The Poincaré operators: Definition

Let D ⊂ R3 be star-shaped with respect to a ∈ D R∇

a u(x)

= (x − a) · 1 u

• a + t(x − a)
• dt=

x

a

u · ds Rcurl

a

u(x) = −(x − a) × 1 t u

• a + t(x − a)
• dt

Rdiv

a u(x)

= (x − a) 1 t2 u

• a + t(x − a)
• dt

Known:

• 1. Polynomials are mapped to polynomials:

Pp Rdiv

a

− → RTp

Raviart-Thomas Rcurl

a

− → W p+1

Nedelec R∇

a

− → Pp+1 Pp

div

← − RTp

curl

← − W p+1

∇

← − Pp+1

• 2. Homotopy relations:

R∇

a ∇u = u − u(a)

Rcurl

a

curl u + ∇R∇

a u = u

Rdiv

a div u + curl Rcurl a

u = u div Rdiv

a u = u

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The Poincaré operators: Definition

Regularized Poincaré operator: θ ∈ C∞

0 (B), D star-shaped with respect to B,

• θ(a)da = 1

Rcurl u(x) = −

• B

θ(a)(x − a) × 1 t u

• a + t(x − a)
• dt da

Rdivu(x) =

• B

θ(a)(x − a) · 1 t2 u

• a + t(x − a)
• dt da

and for differential ℓ-forms u Rℓu(x) =

• B

θ(a)(x − a) 1 tℓ−1 u

• a + t(x − a)
• dt da

Change of variables... Weakly singular kernel Rcurl u(x) = ∞

• r 2 x − y

|x − y|3 + r x − y |x − y|2

• θ
• y − r x − y

|x − y|

• dr × u(y) dy

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Bogovski˘ ı and Poincaré operators. Support properties

Regularized Poincaré operator: Rℓu(x) =

• B

θ(a)(x − a) 1 tℓ−1 u

• a + t(x − a)
• dt da

Bogovski˘ ı integral operator: Tℓu(x) = −

• B

θ(a)(x − a) ∞

1

tℓ−1 u

• a + t(x − a)
• dt da

Duality: Tℓ = ⋆ (Rn−ℓ+1)′ ⋆ Support properties:

• For x ∈ D, Rℓu(x) depends only on u
• D
• If u = 0 on Rd \ D, then Tℓu = 0 on Rd \ D.

Poincaré

Bogovskii

x x

a

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Bogovski˘ ı and Poincaré operators. Main properties Theorem [Costabel-McIntosh 2010] Let D be star-shaped with respect to B. Then ∗ Rℓ, Tℓ are pseudodifferential operators

• f order -1 on Rd

∗ Rℓ maps polynomials to polynomials ∗ dℓ−1Rℓu + Rℓ+1dℓu = u ∗ dℓ−1Tℓu + Tℓ+1dℓu = u ∗ Rℓ : Hs(D, Λℓ) → Hs+1(D, Λℓ−1) ∀s ∈ R ∗ Tℓ : Hs(D, Λℓ) → Hs+1(D, Λℓ−1) ∀s ∈ R

Many other spaces possible: W s,p, Bs

p,q (Besov), F s p,q (Triebel-Lizorkin)

Many other applications known. In particular, our Lemma follows, hence discrete compactness for the p version edge elements in [BuCoDaDeHi 2011].

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Thank you for your attention!

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