Finite element methods for Maxwells equations: A local a priori - - PowerPoint PPT Presentation

• 5. Weihnachtskolloquium

Dec 20, 2016

Finite element methods for Maxwell’s equations: A local a priori estimate

Claudio Rojik

Vienna University of Technology Institute for Analysis and Scientific Computing

Finite element methods for Maxwell’s equations: A local a priori estimate

Motivation

Claudio Rojik (TU Wien)

Finite element methods for Maxwell’s equations: A local a priori estimate Motivation

A local a priori estimate for Poisson’s equation

Assumptions

Ω ⊆ Rn . . . bounded domain

Claudio Rojik (TU Wien) – 1 –

Finite element methods for Maxwell’s equations: A local a priori estimate Motivation

A local a priori estimate for Poisson’s equation

Assumptions

Ω ⊆ Rn . . . bounded domain Poisson’s equation −∆u = f in Ω corresponding bilinear form a(u, v) :=

• Ω ∇u · ∇v dx for u, v ∈ H1(Ω)

Claudio Rojik (TU Wien) – 1 –

Finite element methods for Maxwell’s equations: A local a priori estimate Motivation

A local a priori estimate for Poisson’s equation

Assumptions

Ω ⊆ Rn . . . bounded domain Poisson’s equation −∆u = f in Ω corresponding bilinear form a(u, v) :=

• Ω ∇u · ∇v dx for u, v ∈ H1(Ω)

Th . . . triangulation of Ω with element diameter h Wh(Ω) ⊆ H1(Ω) . . . H1-conforming finite element space

Claudio Rojik (TU Wien) – 1 –

Finite element methods for Maxwell’s equations: A local a priori estimate Motivation

A local a priori estimate for Poisson’s equation

Theorem 1 (Nitsche and Schatz 1974)

Let B0 be a ball with radius r and Bd be the concentric ball with radius r + d, where d ≫ h. (Of course Bd ⊆ Ω must hold.) If uh ∈ Wh(Ω) is a FEM-approximation to u ∈ H1(Ω) satisfying a(u − uh, χ) = 0 for χ ∈ W comp

h

(Bd), then u − uhH1(B0) ≤C min

χ∈Wh(Bd)

• u − χH1(Bd) + d−1u − χL2(Bd)
• + Cd−1u − uhL2(Bd).

Claudio Rojik (TU Wien) – 2 –

Finite element methods for Maxwell’s equations: A local a priori estimate Motivation

Can we establish a similar a priori estimate if we consider Maxwell’s equations instead of Poisson’s equation?

Claudio Rojik (TU Wien) – 3 –

Finite element methods for Maxwell’s equations: A local a priori estimate

FEM setting for Maxwell’s equations

Claudio Rojik (TU Wien)

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations

Maxwell’s equations

E, H . . . electric field intensity, magnetic field intensity B, D . . . magnetic flux density, displacement current density j, ρ . . . electric current density, charge density

Claudio Rojik (TU Wien) – 4 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations

Maxwell’s equations

E, H . . . electric field intensity, magnetic field intensity B, D . . . magnetic flux density, displacement current density j, ρ . . . electric current density, charge density

Maxwell’s equations

curl E = −∂B ∂t curl H = ∂D ∂t + j div D = ρ div B = 0

Claudio Rojik (TU Wien) – 4 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations

Maxwell’s equations

Introduce vector potential A: E = −∂A ∂t Maxwell’s equations + material laws B = µH, j = σE, D = ǫE ⇒ curl(µ−1 curl A) + κA = j µ is the permeability, κ ∈ C is a constant depending on the setting! We will set µ ≡ 1 for simplicity.

Claudio Rojik (TU Wien) – 5 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations

Variational framework

Assumptions

Ω ⊆ R3 bounded domain with polyhedral boundary div j = 0 κ ∈ C\{0}

Claudio Rojik (TU Wien) – 6 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations

Variational framework

Assumptions

Ω ⊆ R3 bounded domain with polyhedral boundary div j = 0 κ ∈ C\{0}

Weak formulation

Find u ∈ H0(curl, Ω) such that b(u, v) = (j, v)L2(Ω) ∀v ∈ H0(curl, Ω) where b(u, v) := (curl u, curl v)L2(Ω) + κ(u, v)L2(Ω).

Claudio Rojik (TU Wien) – 6 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations

Finite element spaces: H1-conforming

Th . . . triangulation of Ω with element diameter h

Claudio Rojik (TU Wien) – 7 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations

Finite element spaces: H1-conforming

Th . . . triangulation of Ω with element diameter h P1(K) denotes the space of polynomials of degree 1 in K

Claudio Rojik (TU Wien) – 7 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations

Finite element spaces: H1-conforming

Th . . . triangulation of Ω with element diameter h P1(K) denotes the space of polynomials of degree 1 in K

Definition

We define Wh(Ω) := {uh ∈ H1(Ω) : (uh)|K ∈ P1(K)∀K ∈ Th}. The space Wh is a H1-conforming finite element space.

Claudio Rojik (TU Wien) – 7 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations

Finite element spaces: H(curl)-conforming

Definition

Using Rk := (Pk−1)3 ⊕ {p ∈ ( ˜ Pk)3 : x · p = 0}, we define Vh(Ω) := {vh ∈ H(curl, Ω) : vh|K ∈ R1∀K ∈ Th}. The space Vh is now H(curl)-conforming.

Claudio Rojik (TU Wien) – 8 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations

Finite element spaces: H(div)-conforming

Definition

Using Dk := (Pk−1)3 ⊕ ˜ Pk−1x we define Qh(Ω) := {qh ∈ H(div, Ω) : qh|K ∈ D1∀K ∈ Th} The space Qh is a H(div)-conforming finite element space.

Claudio Rojik (TU Wien) – 9 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations

The DeRham-diagram

H1(Ω)

∇

− → H(curl, Ω)

curl

− → H(div, Ω)

div

− → L2(Ω) ↓ πh ↓ rh ↓ wh ↓ p0,h Wh(Ω)

∇

− → Vh(Ω)

curl

− → Qh(Ω)

div

− → Sh(Ω)

Claudio Rojik (TU Wien) – 10 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations

An idea of the a priori estimate

We are currently working on establishing an estimate similar to the following:

Theorem 2 (Idea)

Let B0 be a ball with radius r and Bd be the concentric ball with radius r + d, where h ≪ d ≤ 1. (Of course Bd ⊆ Ω must hold.) If uh ∈ Vh(Ω) is a FEM-approximation to u ∈ H(curl, Ω) satisfying b(u − uh, χ) = 0 for χ ∈ V comp

h

(Bd), then u − uhH(curl,B0) ≤C min

χ∈Vh(Bd)

• d−1/2u − χH(curl,Bd)

+d−1u − χL2(Bd)

• + Cd−1u − uhL2(Bd).

Claudio Rojik (TU Wien) – 11 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations

An idea of the a priori estimate

Questions and problems:

Claudio Rojik (TU Wien) – 12 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations

An idea of the a priori estimate

Questions and problems: The bilinear form b contains an L2-term with a constant κ. Change assumptions in the theorem suitably?

Claudio Rojik (TU Wien) – 12 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations

An idea of the a priori estimate

Questions and problems: The bilinear form b contains an L2-term with a constant κ. Change assumptions in the theorem suitably? Proof of Theorem 1 → uh ∈ W comp

h

(Bd) and a(˜ u − uh, uh) = 0 = ⇒ | uh|H1(Bd) ≤ |˜ u|H1(Bd)

Claudio Rojik (TU Wien) – 12 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations

An idea of the a priori estimate

Questions and problems: The bilinear form b contains an L2-term with a constant κ. Change assumptions in the theorem suitably? Proof of Theorem 1 → uh ∈ W comp

h

(Bd) and a(˜ u − uh, uh) = 0 = ⇒ | uh|H1(Bd) ≤ |˜ u|H1(Bd) b(˜ u − uh, uh) = 0 ⇒ ?

Claudio Rojik (TU Wien) – 12 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations

An idea of the a priori estimate

Questions and problems: The bilinear form b contains an L2-term with a constant κ. Change assumptions in the theorem suitably? Proof of Theorem 1 → uh ∈ W comp

h

(Bd) and a(˜ u − uh, uh) = 0 = ⇒ | uh|H1(Bd) ≤ |˜ u|H1(Bd) b(˜ u − uh, uh) = 0 ⇒ ? d−1/2u − χH(curl,Bd) → only quasi-optimality dependent on d. Improvement possible?

Claudio Rojik (TU Wien) – 12 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations

An idea of the a priori estimate

Questions and problems: The bilinear form b contains an L2-term with a constant κ. Change assumptions in the theorem suitably? Proof of Theorem 1 → uh ∈ W comp

h

(Bd) and a(˜ u − uh, uh) = 0 = ⇒ | uh|H1(Bd) ≤ |˜ u|H1(Bd) b(˜ u − uh, uh) = 0 ⇒ ? d−1/2u − χH(curl,Bd) → only quasi-optimality dependent on d. Improvement possible? ”slush term” d−1u − uhL2(Bd) → not ideal in the H(curl)-setting! Reason: L2-norm has same order as H(curl)-norm on gradient subspace

Claudio Rojik (TU Wien) – 12 –

Finite element methods for Maxwell’s equations: A local a priori estimate

The proof of the theorem

Claudio Rojik (TU Wien)

Finite element methods for Maxwell’s equations: A local a priori estimate The proof of the theorem

The proof

Let ω ∈ C∞

0 (Bd) be a cut-off-function with the properties

ω|Bd/2 ≡ 1 and ωW l,∞(Bd) ≤ C1d−l, l = 0, 1, 2. Set ˜ u := ωu, uh . . . Galerkin projection into V comp

h

(Bd), i.e. b(˜ u − uh, χ) = 0 ∀χ ∈ V comp

h

(Bd).

Claudio Rojik (TU Wien) – 13 –

Finite element methods for Maxwell’s equations: A local a priori estimate The proof of the theorem

The proof

Since b( uh − uh, v) = b( uh − ˜ u, v) + b(˜ u − u, v) + b(u − uh, v) = 0 for all v ∈ V comp

h

(Bd/2), we use following inverse estimate with vh = uh − uh:

Lemma

Let h ≪ d ≤ C(κ). If vh ∈ Vh(Ω) satisfies b(vh, χ) = 0 for all χ ∈ V comp

h

(Bd/2), then curl vhL2(B0) d−1vhL2(Bd/2).

Claudio Rojik (TU Wien) – 14 –

Finite element methods for Maxwell’s equations: A local a priori estimate The proof of the theorem

The proof

So it follows u − uhH(curl,B0) ≤ ˜ u − uhH(curl,B0) + uh − uhH(curl,B0) ˜ u − uhH(curl,B0) + d−1 uh − ˜ uL2(Bd) + d−1u − uhL2(Bd) d−1 uh − ˜ uL2(Bd) + curl( uh − ˜ u)L2(Bd) + d−1u − uhL2(Bd) d−1 uhL2(Bd) + d−1uL2(Bd) + curl uL2(Bd) + curl uhL2(Bd) + d−1u − uhL2(Bd).

Claudio Rojik (TU Wien) – 15 –

Finite element methods for Maxwell’s equations: A local a priori estimate The proof of the theorem

The proof

So it follows u − uhH(curl,B0) ≤ ˜ u − uhH(curl,B0) + uh − uhH(curl,B0) ˜ u − uhH(curl,B0) + d−1 uh − ˜ uL2(Bd) + d−1u − uhL2(Bd) d−1 uh − ˜ uL2(Bd) + curl( uh − ˜ u)L2(Bd) + d−1u − uhL2(Bd) d−1 uhL2(Bd) + d−1uL2(Bd) + curl uL2(Bd) + curl uhL2(Bd) + d−1u − uhL2(Bd). The green terms are already as desired, but we have to further estimate the red expressions.

Claudio Rojik (TU Wien) – 15 –

Finite element methods for Maxwell’s equations: A local a priori estimate The proof of the theorem

The proof

Proposition (Friedrichs inequality)

Under the conditions of the theorem, we have

• uhL2(Bd) d

1 2 curl

uhL2(Bd) + ˜ uL2(Bd) and curl uhL2(Bd) curl ˜ uL2(Bd) + d−1˜ uL2(Bd).

Claudio Rojik (TU Wien) – 16 –

Finite element methods for Maxwell’s equations: A local a priori estimate The proof of the theorem

The proof

Proposition (Friedrichs inequality)

Under the conditions of the theorem, we have

• uhL2(Bd) d

1 2 curl

uhL2(Bd) + ˜ uL2(Bd) and curl uhL2(Bd) curl ˜ uL2(Bd) + d−1˜ uL2(Bd). H1-conforming case: More or less Poincar´ e’s inequality!

Claudio Rojik (TU Wien) – 16 –

Finite element methods for Maxwell’s equations: A local a priori estimate The proof of the theorem

The proof

Assuming the validity of the Proposition, we can write: u − uhH(curl,B0) d− 1

2 curl uL2(Bd) + d−1uL2(Bd)

+ d−1u − uhL2(Bd) and u − uh = (u − χ) − (uh − χ) with arbitrary χ ∈ Vh(Bd) implies u − uhH(curl,B0) ≤C min

χ∈Vh(Bd)

• d− 1

2 u − χH(curl,Bd)

+d−1u − χL2(Bd)

• + Cd−1u − uhL2(Bd).

Claudio Rojik (TU Wien) – 17 –

Finite element methods for Maxwell’s equations: A local a priori estimate The proof of the theorem

The proof

Open questions:

Claudio Rojik (TU Wien) – 18 –

Finite element methods for Maxwell’s equations: A local a priori estimate The proof of the theorem

The proof

Open questions: (i) The second inequality of the Proposition follows from the first one. However, how can this line be proved?

Claudio Rojik (TU Wien) – 18 –

Finite element methods for Maxwell’s equations: A local a priori estimate The proof of the theorem

The proof

Open questions: (i) The second inequality of the Proposition follows from the first one. However, how can this line be proved? (ii) Can we get rid of the (red) factor d− 1

2 ? Claudio Rojik (TU Wien) – 18 –

Finite element methods for Maxwell’s equations: A local a priori estimate The proof of the theorem

The proof

Open questions: (i) The second inequality of the Proposition follows from the first one. However, how can this line be proved? (ii) Can we get rid of the (red) factor d− 1

2 ?

(iii) What can we do with the slush term?

Claudio Rojik (TU Wien) – 18 –

Finite element methods for Maxwell’s equations: A local a priori estimate The proof of the theorem

The proof

Open questions: (i) The second inequality of the Proposition follows from the first one. However, how can this line be proved? (ii) Can we get rid of the (red) factor d− 1

2 ?

(iii) What can we do with the slush term? Idea: Helmholtz decomposition, L2-norm for divergence-free part, negative Sobolev norms for gradient subspace.

Claudio Rojik (TU Wien) – 18 –

Finite element methods for Maxwell’s equations: A local a priori estimate The proof of the theorem

The proof

Open questions: (i) The second inequality of the Proposition follows from the first one. However, how can this line be proved? (ii) Can we get rid of the (red) factor d− 1

2 ?

(iii) What can we do with the slush term? Idea: Helmholtz decomposition, L2-norm for divergence-free part, negative Sobolev norms for gradient subspace. → Outline of the proof of (i)

Claudio Rojik (TU Wien) – 18 –

Finite element methods for Maxwell’s equations: A local a priori estimate The proof of the theorem

The Helmholtz decomposition

The following decomposition holds:

• uh = φh + ∇ψh,

φh ∈ V comp

h

(Bd), ψh ∈ W comp

h

(Bd), where φh is discrete divergence free, i.e. (φh, ∇wh)L2(Bd) = 0 ∀wh ∈ W comp

h

(Bd).

Claudio Rojik (TU Wien) – 19 –

Finite element methods for Maxwell’s equations: A local a priori estimate The proof of the theorem

The Helmholtz decomposition

The following decomposition holds:

• uh = φh + ∇ψh,

φh ∈ V comp

h

(Bd), ψh ∈ W comp

h

(Bd), where φh is discrete divergence free, i.e. (φh, ∇wh)L2(Bd) = 0 ∀wh ∈ W comp

h

(Bd). It follows

• uhL2(Bd) ≤ φhL2(Bd) + ∇ψhL2(Bd)

≤ φhL2(Bd) + ˜ uL2(Bd).

Claudio Rojik (TU Wien) – 19 –

Finite element methods for Maxwell’s equations: A local a priori estimate The proof of the theorem

Discrete divergence free functions

Definition

We define Ω := K ∈ Th : K ∩ Bd

c = ∅

• .

Idea: Find a function u ∈ H0(curl, Ω) satisfying curl u = curl φh, (u, ∇φ)L2(Ω) = 0 ∀φ ∈ H1

0(Ω)

and u − φhL2(Ω) h

1 2 curl φhL2(Ω). Claudio Rojik (TU Wien) – 20 –

Finite element methods for Maxwell’s equations: A local a priori estimate The proof of the theorem

Discrete divergence free functions

Ω . . . smooth boundary or convex domain ⇒ u − uhL2(Ω) h curl uhL2(Ω) due to regularity theory of elliptic PDEs.

Claudio Rojik (TU Wien) – 21 –

Finite element methods for Maxwell’s equations: A local a priori estimate The proof of the theorem

Discrete divergence free functions

Ω . . . smooth boundary or convex domain ⇒ u − uhL2(Ω) h curl uhL2(Ω) due to regularity theory of elliptic PDEs.

Problem

Claudio Rojik (TU Wien) – 21 –

Finite element methods for Maxwell’s equations: A local a priori estimate The proof of the theorem

Discrete divergence free functions

Ω . . . smooth boundary or convex domain ⇒ u − uhL2(Ω) h curl uhL2(Ω) due to regularity theory of elliptic PDEs.

Problem

In general, the domain Ω doesn’t even have a Lipschitz boundary!

Claudio Rojik (TU Wien) – 21 –

Finite element methods for Maxwell’s equations: A local a priori estimate

Divergence free functions in star-shaped domains

Claudio Rojik (TU Wien)

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Star-shaped Lipschitz domains

Since we need some theorems that only hold on Lipschitz domains, we enforce a Lipschitz condition on the boundary by restrictions on the mesh!

Claudio Rojik (TU Wien) – 22 –

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Star-shaped Lipschitz domains

Since we need some theorems that only hold on Lipschitz domains, we enforce a Lipschitz condition on the boundary by restrictions on the mesh!

Lemma

Let ch ≤ d for c ≫ 1, and let arccos( 1

2c) − δ be the maximal dihedral

angle of every tetrahedral element of Th, where 0 < δ ≪ 1 is sufficiently small.

Claudio Rojik (TU Wien) – 22 –

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Star-shaped Lipschitz domains

Since we need some theorems that only hold on Lipschitz domains, we enforce a Lipschitz condition on the boundary by restrictions on the mesh!

Lemma

Let ch ≤ d for c ≫ 1, and let arccos( 1

2c) − δ be the maximal dihedral

angle of every tetrahedral element of Th, where 0 < δ ≪ 1 is sufficiently small. Then there exists a ˆ C = ˆ C(δ) such that Ω ⊆ Bd is star-shaped with respect to the ball B ˆ

Cd.

Claudio Rojik (TU Wien) – 22 –

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Star-shaped Lipschitz domains

Since we need some theorems that only hold on Lipschitz domains, we enforce a Lipschitz condition on the boundary by restrictions on the mesh!

Lemma

Let ch ≤ d for c ≫ 1, and let arccos( 1

2c) − δ be the maximal dihedral

angle of every tetrahedral element of Th, where 0 < δ ≪ 1 is sufficiently small. Then there exists a ˆ C = ˆ C(δ) such that Ω ⊆ Bd is star-shaped with respect to the ball B ˆ

Cd.

The domain Ω is a Lipschitz domain with Lipschitz constant comparable to γ. γ =

d ρmax ∼ ˆ

C−1 is called the chunkiness parameter of Ω.

Claudio Rojik (TU Wien) – 22 –

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Finding the divergence free function

Remember:

Proposition

φh ∈ Vh(Ω) discrete divergence free, tangential trace zero on the boundary ⇒ There exists a unique divergence free function u ∈ H0(curl, Ω) with same curl, satisfying u − φhL2(Ω) h

1 2 curl φhL2(Ω).

Idea of the proof: Define u by a saddle point problem.

Claudio Rojik (TU Wien) – 23 –

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Saddle point problems

Theorem (Brezzi)

Saddle point problem: Find u ∈ V and p ∈ Q such that a(u, v) + b(v, p) = f(v) ∀v ∈ V b(u, q) = g(q) ∀q ∈ Q. . If (coercivity on the kernel) a(v, v) ≥ αv2

V

∀v ∈ {v ∈ V : b(v, q) = 0 ∀q ∈ Q} and (LBB-condition) inf

q∈Q sup v∈V

b(v, q) vV qQ ≥ β ⇒ unique solution.

Claudio Rojik (TU Wien) – 24 –

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Concept of the proof

Saddle point problem

Find u ∈ H0(curl, Ω) and p ∈ curl (H0(curl, Ω)) ⊆ H0(div, Ω) such that (u, v)L2(Ω) + (curl v, p)L2(Ω) = (curl u, q)L2(Ω) = (curl φh, q)L2(Ω) ∀v ∈ H0(curl, Ω) ∀q ∈ curl (H0(curl, Ω)).

Claudio Rojik (TU Wien) – 25 –

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Concept of the proof

Saddle point problem

Find u ∈ H0(curl, Ω) and p ∈ curl (H0(curl, Ω)) ⊆ H0(div, Ω) such that (u, v)L2(Ω) + (curl v, p)L2(Ω) = (curl u, q)L2(Ω) = (curl φh, q)L2(Ω) ∀v ∈ H0(curl, Ω) ∀q ∈ curl (H0(curl, Ω)). coercivity on the kernel:

Claudio Rojik (TU Wien) – 25 –

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Concept of the proof

Saddle point problem

Find u ∈ H0(curl, Ω) and p ∈ curl (H0(curl, Ω)) ⊆ H0(div, Ω) such that (u, v)L2(Ω) + (curl v, p)L2(Ω) = (curl u, q)L2(Ω) = (curl φh, q)L2(Ω) ∀v ∈ H0(curl, Ω) ∀q ∈ curl (H0(curl, Ω)). coercivity on the kernel: (trivial)

Claudio Rojik (TU Wien) – 25 –

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Concept of the proof

Saddle point problem

Find u ∈ H0(curl, Ω) and p ∈ curl (H0(curl, Ω)) ⊆ H0(div, Ω) such that (u, v)L2(Ω) + (curl v, p)L2(Ω) = (curl u, q)L2(Ω) = (curl φh, q)L2(Ω) ∀v ∈ H0(curl, Ω) ∀q ∈ curl (H0(curl, Ω)). coercivity on the kernel: (trivial) LBB-condition:

Claudio Rojik (TU Wien) – 25 –

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Concept of the proof

Saddle point problem

Find u ∈ H0(curl, Ω) and p ∈ curl (H0(curl, Ω)) ⊆ H0(div, Ω) such that (u, v)L2(Ω) + (curl v, p)L2(Ω) = (curl u, q)L2(Ω) = (curl φh, q)L2(Ω) ∀v ∈ H0(curl, Ω) ∀q ∈ curl (H0(curl, Ω)). coercivity on the kernel: (trivial) LBB-condition: (non-trivial!)

Claudio Rojik (TU Wien) – 25 –

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Concept of the proof

Saddle point problem

Find u ∈ H0(curl, Ω) and p ∈ curl (H0(curl, Ω)) ⊆ H0(div, Ω) such that (u, v)L2(Ω) + (curl v, p)L2(Ω) = (curl u, q)L2(Ω) = (curl φh, q)L2(Ω) ∀v ∈ H0(curl, Ω) ∀q ∈ curl (H0(curl, Ω)). coercivity on the kernel: (trivial) LBB-condition: (non-trivial!) ⇒ u ∈ H0(curl, Ω) well-defined.

Claudio Rojik (TU Wien) – 25 –

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Concept of the proof

Saddle point problem

Find u ∈ H0(curl, Ω) and p ∈ curl (H0(curl, Ω)) ⊆ H0(div, Ω) such that (u, v)L2(Ω) + (curl v, p)L2(Ω) = (curl u, q)L2(Ω) = (curl φh, q)L2(Ω) ∀v ∈ H0(curl, Ω) ∀q ∈ curl (H0(curl, Ω)).

Claudio Rojik (TU Wien) – 26 –

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Concept of the proof

Saddle point problem

Find u ∈ H0(curl, Ω) and p ∈ curl (H0(curl, Ω)) ⊆ H0(div, Ω) such that (u, v)L2(Ω) + (curl v, p)L2(Ω) = (curl u, q)L2(Ω) = (curl φh, q)L2(Ω) ∀v ∈ H0(curl, Ω) ∀q ∈ curl (H0(curl, Ω)). v = ∇φ for φ ∈ H1

0(Ω) ⇒ (u, ∇φ)L2(Ω)

= ⇒ u divergence-free

Claudio Rojik (TU Wien) – 26 –

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Concept of the proof

Saddle point problem

Find u ∈ H0(curl, Ω) and p ∈ curl (H0(curl, Ω)) ⊆ H0(div, Ω) such that (u, v)L2(Ω) + (curl v, p)L2(Ω) = (curl u, q)L2(Ω) = (curl φh, q)L2(Ω) ∀v ∈ H0(curl, Ω) ∀q ∈ curl (H0(curl, Ω)). v = ∇φ for φ ∈ H1

0(Ω) ⇒ (u, ∇φ)L2(Ω)

= ⇒ u divergence-free q = curl(u − φh) ⇒ curl u = curl φh

Claudio Rojik (TU Wien) – 26 –

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Concept of the proof

Pose the corresponding finite element saddle point problem:

FEM saddle point problem

Find φh ∈ V0,h(Ω) and ph ∈ curl(V0,h(Ω)) such that ( φh, vh)L2(Ω) + (curl vh, ph)L2(Ω) = (curl φh, qh)L2(Ω) = (curl φh, qh)L2(Ω) ∀vh ∈ V0,h(Ω) ∀qh ∈ curl(V0,h(Ω)).

Claudio Rojik (TU Wien) – 27 –

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Concept of the proof

Pose the corresponding finite element saddle point problem:

FEM saddle point problem

Find φh ∈ V0,h(Ω) and ph ∈ curl(V0,h(Ω)) such that ( φh, vh)L2(Ω) + (curl vh, ph)L2(Ω) = (curl φh, qh)L2(Ω) = (curl φh, qh)L2(Ω) ∀vh ∈ V0,h(Ω) ∀qh ∈ curl(V0,h(Ω)). qh = curl( φh − φh) ⇒ curl( φh − φh) = 0 ⇒ ∃ψh ∈ W0,h(Ω) such that φh − φh = ∇ψh vh = ∇ψh = ⇒ 0 = ( φh, ∇ψh)L2(Ω) − (φh, ∇ψh)L2(Ω) ⇒ φh = φh

Claudio Rojik (TU Wien) – 27 –

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Concept of the proof

Combine the saddle point problems: ⇒ u − φhL2(Ω) ≤ u − rhuL2(Ω)

Claudio Rojik (TU Wien) – 28 –

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Concept of the proof

Combine the saddle point problems: ⇒ u − φhL2(Ω) ≤ u − rhuL2(Ω) Next steps:

Claudio Rojik (TU Wien) – 28 –

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Concept of the proof

Combine the saddle point problems: ⇒ u − φhL2(Ω) ≤ u − rhuL2(Ω) Next steps: Helmholtz decomposition of u

Claudio Rojik (TU Wien) – 28 –

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Concept of the proof

Combine the saddle point problems: ⇒ u − φhL2(Ω) ≤ u − rhuL2(Ω) Next steps: Helmholtz decomposition of u Use definition of rh (degrees of freedom on every edge)

Claudio Rojik (TU Wien) – 28 –

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Concept of the proof

Combine the saddle point problems: ⇒ u − φhL2(Ω) ≤ u − rhuL2(Ω) Next steps: Helmholtz decomposition of u Use definition of rh (degrees of freedom on every edge) Estimates for the edges on the boundary: Theory of harmonic functions (harmonic measure, non-tangential maximal functions, Newton potential . . . )

Claudio Rojik (TU Wien) – 28 –

Finite element methods for Maxwell’s equations: A local a priori estimate Divergence free functions in star-shaped domains

Concept of the proof

Combine the saddle point problems: ⇒ u − φhL2(Ω) ≤ u − rhuL2(Ω) Next steps: Helmholtz decomposition of u Use definition of rh (degrees of freedom on every edge) Estimates for the edges on the boundary: Theory of harmonic functions (harmonic measure, non-tangential maximal functions, Newton potential . . . ) → still not finished, but the estimate u − φhL2(Ω) h

1 2 curl φhL2(Ω)

looks possible!

Claudio Rojik (TU Wien) – 28 –

Finite element methods for Maxwell’s equations: A local a priori estimate

Summary

Claudio Rojik (TU Wien)

Finite element methods for Maxwell’s equations: A local a priori estimate Summary

Summary

Theorem

Assume: Triangulation Th with maximal angle π

2 − δ for every K ∈ Th

B0 . . . ball, radius r, Bd . . . concentric ball, radius r + d, d ≫ h uh ∈ Vh approx. to u ∈ H(curl) with b(u − uh, χ) = 0, χ ∈ V comp

h

= ⇒ u − uhH(curl,B0) ≤C min

χ∈Vh(Bd)

• d−1/2u − χH(curl,Bd)

+ d−1u − χL2(Bd)

• + Cd−1u − uhL2(Bd)

Claudio Rojik (TU Wien) – 29 –

Finite element methods for Maxwell’s equations: A local a priori estimate Summary

Summary

Theorem

Assume: Triangulation Th with maximal angle π

2 − δ for every K ∈ Th

B0 . . . ball, radius r, Bd . . . concentric ball, radius r + d, d ≫ h uh ∈ Vh approx. to u ∈ H(curl) with b(u − uh, χ) = 0, χ ∈ V comp

h

= ⇒ u − uhH(curl,B0) ≤C min

χ∈Vh(Bd)

• d−1/2u − χH(curl,Bd)

+ d−1u − χL2(Bd)

• + Cd−1u − uhL2(Bd)

Unfinished parts of the proof:

Claudio Rojik (TU Wien) – 29 –

Finite element methods for Maxwell’s equations: A local a priori estimate Summary

Summary

Theorem

Assume: Triangulation Th with maximal angle π

2 − δ for every K ∈ Th

B0 . . . ball, radius r, Bd . . . concentric ball, radius r + d, d ≫ h uh ∈ Vh approx. to u ∈ H(curl) with b(u − uh, χ) = 0, χ ∈ V comp

h

= ⇒ u − uhH(curl,B0) ≤C min

χ∈Vh(Bd)

• d−1/2u − χH(curl,Bd)

+ d−1u − χL2(Bd)

• + Cd−1u − uhL2(Bd)

Unfinished parts of the proof: Friedrichs inequality (i.e. last estimate between divergence-free and discrete divergence-free function)

Claudio Rojik (TU Wien) – 29 –

Finite element methods for Maxwell’s equations: A local a priori estimate Summary

Summary

Theorem

Assume: Triangulation Th with maximal angle π

2 − δ for every K ∈ Th

B0 . . . ball, radius r, Bd . . . concentric ball, radius r + d, d ≫ h uh ∈ Vh approx. to u ∈ H(curl) with b(u − uh, χ) = 0, χ ∈ V comp

h

= ⇒ u − uhH(curl,B0) ≤C min

χ∈Vh(Bd)

• d−1/2u − χH(curl,Bd)

+ d−1u − χL2(Bd)

• + Cd−1u − uhL2(Bd)

Unfinished parts of the proof: Friedrichs inequality (i.e. last estimate between divergence-free and discrete divergence-free function) weaker norm for the slush term to get improved convergence

Claudio Rojik (TU Wien) – 29 –

Finite element methods for Maxwell’s equations: A local a priori estimate Summary

Thank you for your attention!

Claudio Rojik (TU Wien) – 30 –