Théophile Chaumont-Frelet, Alexandre Ern, Martin Vohralík

Polynomial-degree-robust a posteriori error estimationfor the curl-curl problem

Inria Paris & Ecole des Ponts

RANAPDE, June 24, 2021

I H1-case H(curl)-case Equilibration Polynomial extensions Numerics C

Outline

1 Introduction

2 Reminder on the H1-case

3 The H(curl)-case

4 H(curl) patchwise equilibration

5 Stable (broken) H(curl) polynomial extensions

6 Numerical experiments

7 Conclusions

M. Vohralík p-robust a posteriori estimation for the curl-curl problem 1 / 29

The curl-curl problem (current density j ∈ H0,N(div,Ω) with ∇·j = 0)

The curl–curl problem

Find the magnetic vector potential A : Ω→ R3 such that∇×(∇×A) = j , ∇·A = 0 in Ω,

A×nΩ = 0, on ΓD,

(∇×A)×nΩ = 0, A·nΩ = 0 on ΓN.

Weak formulation (consequence)

A ∈ H0,D(curl,Ω) satisfies(∇×A,∇×v) = (j ,v) ∀v ∈ H0,D(curl,Ω).

Nédélec finite element discretization (consequence)

V h := Np(Th) ∩ H0,D(curl,Ω), p ≥ 0; Ah ∈ V h satisfies(∇×Ah,∇×vh) = (j ,vh) ∀vh ∈ V h.

A×nΩ = 0, on ΓD,

(∇×A)×nΩ = 0, A·nΩ = 0 on ΓN.

A×nΩ = 0, on ΓD,

(∇×A)×nΩ = 0, A·nΩ = 0 on ΓN.

BibliographyResidual estimates

Monk (1998)Beck, Hiptmair, Hoppe, & Wohlmuth (2000)Nicaise & Creusé (2003)

Functional estimatesRepin (2007)Hannukainen (2008)Neittaanmäki & Repin (2010)

Equlibrated estimatesBraess & Schöberl (2008): patchwise minimizations (lowest-order case p = 0)Licht (2019): a conceptual discussionGedicke, Geevers, & Perugia (2020): equilibrated-residual-style constructionGedicke, Geevers, Perugia, & Schöberl (2020): p-robust modification

Outline

1 Introduction

3 The H(curl)-case

7 Conclusions

The hat function and the partition of unity, Ω ⊂ Rd

−0.50

0.51−1

−0.5

The hat function ψa, d = 2

Partition of unity

∑a∈Vh

ψa = 1|Ω

The Laplacian −∆u = f in Ω, u = 0 on ∂Ω

Weak solution u ∈ H10 (Ω) is such that

(∇u,∇v) = (f , v) ∀v ∈ H10 (Ω)

Approximation uh ∈ H10 (Ω) satisfies

(∇uh,∇ψa) = (f , ψa) ∀a ∈ V inth

Residual R(uh) ∈ H−1(Ω) is defined by〈R(uh), v〉 := (f , v)− (∇uh,∇v)

Norm characterization‖∇(u−uh)‖ = ‖R(uh)‖−1 = sup

v∈H10 (Ω)

‖∇v‖=1

〈R(uh), v〉

∗ (ωa) :=

v ∈ H1(ωa); (v ,1)ωa = 0 for interior vertex a ∈ V int

v ∈ H1(ωa); v = 0 on faces sharing a for boundary vertex a ∈ Vexth

ψa-weighted residual on H1∗ (ωa)′

‖∇(u − uh)‖ ≤ (d + 1)1/2∑a∈Vh

supv∈H1

∗(ωa)‖∇v‖ωa =1

〈R(uh), ψav〉

Unweighted residual on H10 (ωa)′

‖∇(u − uh)‖ ≤ (d + 1)1/2Ccont,PF∑a∈Vh

supv∈H1

0 (ωa)‖∇v‖ωa =1

〈R(uh), v〉

(∇u,∇v) = (f , v) ∀v ∈ H10 (Ω)

v∈H10 (Ω)

‖∇v‖=1

〈R(uh), v〉

∗ (ωa) :=

‖∇(u − uh)‖ ≤ (d + 1)1/2∑a∈Vh

supv∈H1

∗(ωa)‖∇v‖ωa =1

〈R(uh), ψav〉

‖∇(u − uh)‖ ≤ (d + 1)1/2Ccont,PF∑a∈Vh

supv∈H1

0 (ωa)‖∇v‖ωa =1

〈R(uh), v〉

(∇u,∇v) = (f , v) ∀v ∈ H10 (Ω)

v∈H10 (Ω)

‖∇v‖=1

〈R(uh), v〉

∗ (ωa) :=

‖∇(u − uh)‖ ≤ (d + 1)1/2∑a∈Vh

supv∈H1

∗(ωa)‖∇v‖ωa =1

〈R(uh), ψav〉

‖∇(u − uh)‖ ≤ (d + 1)1/2Ccont,PF∑a∈Vh

supv∈H1

0 (ωa)‖∇v‖ωa =1

〈R(uh), v〉

(∇u,∇v) = (f , v) ∀v ∈ H10 (Ω)

v∈H10 (Ω)

‖∇v‖=1

〈R(uh), v〉

∗ (ωa) :=

‖∇(u − uh)‖ ≤ (d + 1)1/2∑a∈Vh

supv∈H1

∗(ωa)‖∇v‖ωa =1

〈R(uh), ψav〉

‖∇(u − uh)‖ ≤ (d + 1)1/2Ccont,PF∑a∈Vh

supv∈H1

0 (ωa)‖∇v‖ωa =1

〈R(uh), v〉

(∇u,∇v) = (f , v) ∀v ∈ H10 (Ω)

v∈H10 (Ω)

‖∇v‖=1

〈R(uh), v〉

∗ (ωa) :=

‖∇(u − uh)‖ ≤ (d + 1)1/2∑a∈Vh

supv∈H1

∗(ωa)‖∇v‖ωa =1

〈R(uh), ψav〉

‖∇(u − uh)‖ ≤ (d + 1)1/2Ccont,PF∑a∈Vh

supv∈H1

0 (ωa)‖∇v‖ωa =1

〈R(uh), v〉

Bound by ψa-weighted residuals on H1∗ (ωa)′

v ∈ H10 (Ω), ‖∇v‖ = 1:

〈R(uh), v〉 PU=⟨R(uh),

∑a∈Vh

(ψav)⟩

GO=∑a∈Vh

⟨R(uh), ψa(v − Π0v)

⟩=∑a∈Vh

〈R(uh), ψa(v − Π0v)〉‖∇v‖ωa

‖∇v‖ωa

∑a∈Vh

supw∈H1

∗(ωa)‖∇w‖ωa =1

〈R(uh), ψaw〉

1/2∑a∈Vh

‖∇v‖2ωa

overlaps≤ (d + 1)1/2

∑a∈Vh

supw∈H1

∗(ωa)‖∇w‖ωa =1

〈R(uh), ψaw〉

Bound by unweighted residuals on H10 (ωa)′

for v ∈ H1∗ (ωa):

‖∇(ψav)‖ωa ≤ ‖∇ψav‖ωa + ‖ψa∇v‖ωa ≤ (1 + CPFhωa‖∇ψa‖∞,ωa )︸︷︷︸≤Ccont,PF

‖∇v‖ωa

bound for the ψa-weighted residual on H1∗ (ωa)′, since ψav ∈ H1

0 (ωa) forv ∈ H1

∗ (ωa):

supv∈H1

∗(ωa)‖∇v‖ωa =1

〈R(uh), ψav〉 = supv∈H1

∗(ωa)

〈R(uh), ψav〉‖∇v‖ωa

= supv∈H1

∗(ωa)

〈R(uh), ψav〉‖∇(ψav)‖ωa

‖∇(ψav)‖ωa

‖∇v‖ωa

≤ Ccont,PF supv∈H1

0 (ωa)‖∇v‖ωa =1

〈R(uh), v〉

Patchwise bounds by equilibrated fluxesψa-weighted residual on H1

∗ (ωa)′

for v ∈ H1∗ (ωa) with ‖∇v‖ωa = 1 and

σah ∈ H(div, ωa) with σa

h·n|∂ωa = 0 on ∂ωa

and ∇·σah = fψa −∇uh·∇ψa,

〈R(uh), ψav〉= (f , ψav)ωa − (∇uh,∇(ψav))ωa

= (fψa −∇uh·∇ψa, v)ωa − (ψa∇uh,∇v)ωa

= (∇·σah, v)ωa − (ψa∇uh,∇v)ωa

Green= − (ψa∇uh + σa

h,∇v)ωaCS≤ ‖ψa∇uh + σa

h‖ωa

supv∈H1

∗(ωa)‖∇v‖ωa =1

〈R(uh), ψav〉 ≤ ‖ψa∇uh + σah‖ωa

for v ∈ H10 (ωa) with ‖∇v‖ωa = 1 and

σah ∈ H(div, ωa) with ∇·σa

h = f ,〈R(uh), v〉 = (f , v)ωa − (∇uh,∇v)ωa

= (∇·σah, v)ωa − (∇uh,∇v)ωa

Green= −(∇uh + σa

h,∇v)ωaCS≤ ‖∇uh + σa

h‖ωa

supv∈H1

0 (ωa)‖∇v‖ωa =1

〈R(uh), v〉 ≤ ‖∇uh + σah‖ωa

∗ (ωa)′

h‖ωa

supv∈H1

∗(ωa)‖∇v‖ωa =1

h‖ωa

supv∈H1

0 (ωa)‖∇v‖ωa =1

∗ (ωa)′

h‖ωa

supv∈H1

∗(ωa)‖∇v‖ωa =1

h‖ωa

supv∈H1

0 (ωa)‖∇v‖ωa =1

∗ (ωa)′

h‖ωa

supv∈H1

∗(ωa)‖∇v‖ωa =1

h‖ωa

supv∈H1

0 (ωa)‖∇v‖ωa =1

∗ (ωa)′

h‖ωa

supv∈H1

∗(ωa)‖∇v‖ωa =1

h‖ωa

supv∈H1

0 (ωa)‖∇v‖ωa =1

∗ (ωa)′

h‖ωa

supv∈H1

∗(ωa)‖∇v‖ωa =1

h‖ωa

supv∈H1

0 (ωa)‖∇v‖ωa =1

∗ (ωa)′

h‖ωa

supv∈H1

∗(ωa)‖∇v‖ωa =1

h‖ωa

supv∈H1

0 (ωa)‖∇v‖ωa =1

Discrete patchwise/broken patchwise equilibrated fluxesDefinition (Destuynder and Métivet (1999) & Braess and Schöberl (2008))For each vertex a ∈ Vh, solve the localconstrained minimization pbσa

h := arg minvh∈RTp+1(Ta)∩H0(div,ωa)∇·vh=Πp+1(fψa−∇uh·∇ψa)

‖ψa∇uh + vh‖2ωa

and combine σh :=∑a∈Vh

a ∈ V int

interior patch ωa

no-flow conditiona1

Key pointshomogeneous normal BC on ∂ωa:σh ∈RTp+1(Th) ∩ H(div,Ω)

global equilibrium ∇·σh =∑a∈Vh

∇·σah

=∑a∈Vh

Πp+1(fψa −∇uh·∇ψa) = Πp+1f

Definition (∼ Chaumont-Frelet, Ern, Vohralík (2021))For each vertex a ∈ Vh, solve the localconstrained minimization pbσa

h := arg minvh∈RTp(Ta)∩H(div,ωa)

∇·vh=Πp f

‖∇uh + vh‖2ωa

Key pointsno BC on ∂ωa:σh =

∑a∈Vh

σah 6∈ H(div,Ω)

only local equilibrium ∇·σah = Πpf

a ∈ V int

interior patch ωa

no-flow conditiona1

∇·σah

=∑a∈Vh

∇·vh=Πp f

∑a∈Vh

σah 6∈ H(div,Ω)

a ∈ V int

interior patch ωa

no-flow conditiona1

∇·σah

=∑a∈Vh

∇·vh=Πp f

∑a∈Vh

σah 6∈ H(div,Ω)

a ∈ V int

interior patch ωa

no-flow conditiona1

∇·σah

=∑a∈Vh

∇·vh=Πp f

∑a∈Vh

σah 6∈ H(div,Ω)

a ∈ V int

interior patch ωa

no-flow conditiona1

∇·σah

=∑a∈Vh

∇·vh=Πp f

∑a∈Vh

σah 6∈ H(div,Ω)

a ∈ V int

interior patch ωa

no-flow conditiona1

∇·σah

=∑a∈Vh

∇·vh=Πp f

a ∈ V int

interior patch ωaa1

∑a∈Vh

σah 6∈ H(div,Ω)

a ∈ V int

interior patch ωa

no-flow conditiona1

∇·σah

=∑a∈Vh

∇·vh=Πp f

a ∈ V int

∑a∈Vh

σah 6∈ H(div,Ω)

a ∈ V int

interior patch ωa

no-flow conditiona1

∇·σah

=∑a∈Vh

∇·vh=Πp f

a ∈ V int

∑a∈Vh

σah 6∈ H(div,Ω)

a ∈ V int

interior patch ωa

no-flow conditiona1

∇·σah

=∑a∈Vh

∇·vh=Πp f

a ∈ V int

∑a∈Vh

σah 6∈ H(div,Ω)

Guaranteed upper bound

‖∇(u − uh)‖︸︷︷︸unknown error

≤ ‖∇uh + σh‖︸︷︷︸computable estimator

≤ (d + 1)1/2

∑a∈Vh

‖ψa∇uh + σah‖2ωa

p-robust local efficiency (Braess, Pillwein, Schöberl (2009;

2D), Ern & V. (2020; 3D))

‖ψa∇uh + σah‖ωa ≤ Cst sup

v∈H1∗(ωa)

‖∇v‖ωa =1

〈R(uh), ψav〉

≤CstCcont,PF supv∈H1

0 (ωa)‖∇v‖ωa =1

〈R(uh), v〉

≤(d + 1)1/2Ccont,PF

∑a∈Vh

‖∇uh + σah‖2ωa

︸︷︷︸computable estimator

p-robust local efficiency (Costabel & Mc-Intosh (2010);

Demkowicz, Gopalakrishnan, & Schöberl (2012))

‖∇uh + σah‖ωa ≤ Cst sup

v∈H10 (ωa)

‖∇v‖ωa =1

〈R(uh), v〉

≤ Cst‖∇(u − uh)‖ωa

≤ (d + 1)1/2

∑a∈Vh

2D), Ern & V. (2020; 3D))

v∈H1∗(ωa)

‖∇v‖ωa =1

〈R(uh), ψav〉

0 (ωa)‖∇v‖ωa =1

〈R(uh), v〉

∑a∈Vh

v∈H10 (ωa)

‖∇v‖ωa =1

〈R(uh), v〉

≤ (d + 1)1/2

∑a∈Vh

2D), Ern & V. (2020; 3D))

v∈H1∗(ωa)

‖∇v‖ωa =1

〈R(uh), ψav〉

0 (ωa)‖∇v‖ωa =1

〈R(uh), v〉

∑a∈Vh

v∈H10 (ωa)

‖∇v‖ωa =1

〈R(uh), v〉

≤ (d + 1)1/2

∑a∈Vh

2D), Ern & V. (2020; 3D))

v∈H1∗(ωa)

‖∇v‖ωa =1

〈R(uh), ψav〉

0 (ωa)‖∇v‖ωa =1

〈R(uh), v〉

∑a∈Vh

v∈H10 (ωa)

‖∇v‖ωa =1

〈R(uh), v〉

≤ (d + 1)1/2

∑a∈Vh

2D), Ern & V. (2020; 3D))

v∈H1∗(ωa)

‖∇v‖ωa =1

〈R(uh), ψav〉

0 (ωa)‖∇v‖ωa =1

〈R(uh), v〉 ≤ ‖∇(u − uh)‖ωa

∑a∈Vh

v∈H10 (ωa)

‖∇v‖ωa =1

〈R(uh), v〉

Outline

1 Introduction

3 The H(curl)-case

7 Conclusions

A×nΩ = 0, on ΓD,

(∇×A)×nΩ = 0, A·nΩ = 0 on ΓN.

A×nΩ = 0, on ΓD,

(∇×A)×nΩ = 0, A·nΩ = 0 on ΓN.

A×nΩ = 0, on ΓD,

(∇×A)×nΩ = 0, A·nΩ = 0 on ΓN.

Discrete patchwise/broken patchwise equilibrated fluxesDefinition (Chaumont-Frelet, Vohralík (2021))For each vertex a ∈ Vh, solve the localconstrained minimization pbha

h := arg minvh∈Np+1(Ta)∩H0(curl,ωa)∇×vh=ψaj+∇ψa×(∇×Ah)

‖ψa(∇×Ah)−vh‖2ωa

and combinehh :=

∑a∈Vh

• a Ta

Key pointshomogeneous tangential BC on ∂ωa:hh ∈Np+1(Th) ∩ H(curl,Ω)

global equilibrium ∇×hh =∑a∈Vh

∇×hah

=∑a∈Vh

(ψaj +∇ψa×(∇×Ah)) = j

Definition (Chaumont-Frelet, Ern, Vohralík (2021))For each edge e ∈ Eh, solve the localconstrained minimization pbhe

h := arg minvh∈Np(Te)∩H(curl,ωe)

∇×vh=j

‖∇×Ah−vh‖2ωe .

Key pointsno BC on ∂ωe:hh =

∑e∈Eh

heh 6∈ H(curl,Ω)

only local equilibrium ∇×heh = j

and combinehh :=

∑a∈Vh

• a Ta

∇×hah

=∑a∈Vh

(ψaj +∇ψa×(∇×Ah)) = j

∇×vh=j

∑e∈Eh

heh 6∈ H(curl,Ω)

and combinehh :=

∑a∈Vh

• a Ta

∇×hah

=∑a∈Vh

(ψaj +∇ψa×(∇×Ah)) = j

∇×vh=j

∑e∈Eh

heh 6∈ H(curl,Ω)

and combinehh :=

∑a∈Vh

• a Ta

∇×hah

=∑a∈Vh

(ψaj +∇ψa×(∇×Ah)) = j

∇×vh=j

∑e∈Eh

heh 6∈ H(curl,Ω)

and combinehh :=

∑a∈Vh

• a Ta

∇×hah

=∑a∈Vh

(ψaj +∇ψa×(∇×Ah)) = j

∇×vh=j

∑e∈Eh

heh 6∈ H(curl,Ω)

and combinehh :=

∑a∈Vh

• a Ta

∇×hah

=∑a∈Vh

(ψaj +∇ψa×(∇×Ah)) = j

∇×vh=j

∑e∈Eh

heh 6∈ H(curl,Ω)

and combinehh :=

∑a∈Vh

• a Ta

∇×hah

=∑a∈Vh

(ψaj +∇ψa×(∇×Ah)) = j

∇×vh=j

∑e∈Eh

heh 6∈ H(curl,Ω)

and combinehh :=

∑a∈Vh

• a Ta

∇×hah

=∑a∈Vh

(ψaj +∇ψa×(∇×Ah)) = j

∇×vh=j

∑e∈Eh

heh 6∈ H(curl,Ω)

and combinehh :=

∑a∈Vh

• a Ta

∇×hah

=∑a∈Vh

(ψaj +∇ψa×(∇×Ah)) = j

∇×vh=j

∑e∈Eh

heh 6∈ H(curl,Ω)

and combinehh :=

∑a∈Vh

• a Ta

∇×hah

=∑a∈Vh

(ψaj +∇ψa×(∇×Ah)) = j

∇×vh=j

well-posed for j|ωe ∈RTp(Te) ∩ H(div, ωe) with∇·j = 0

∑e∈Eh

heh 6∈ H(curl,Ω)

ψa j ∈RTp+1(Ta) ∩ H0(div, ωa) but∇ψa×(∇×Ah) 6∈ H(div, ωa)

hh :=∑a∈Vh

• a Ta

∇×hah

=∑a∈Vh

(ψaj +∇ψa×(∇×Ah)) = j

∇×vh=j

∑e∈Eh

heh 6∈ H(curl,Ω)

ψa j ∈RTp+1(Ta) ∩ H0(div, ωa) but∇ψa×(∇×Ah) 6∈ H(div, ωa)

hh :=∑a∈Vh

• a Ta

∇×hah

=∑a∈Vh

(ψaj +∇ψa×(∇×Ah)) = j

∇×vh=j

∑e∈Eh

heh 6∈ H(curl,Ω)

Bottom line

Continuous caseWhen there exist v ∈ H(curl,Ω) such that ∇×v = j?When j ∈ H(div,Ω) with ∇·j = 0.

Discrete caseWhen there exist vh ∈Np(Th) ∩ H(curl,Ω) such that ∇×vh = j?When j ∈RTp(Th) ∩ H(div,Ω) with ∇·j = 0.

We supposej ∈ H0,N(div,Ω) with ∇·j = 0j ∈RTp(Th) (no data oscillation, simplicity of presentation)

Bottom line

The curl–curl case

‖∇×(A− Ah)‖︸︷︷︸unknown error

≤ ‖∇×Ah − hh‖︸︷︷︸computable estimator

∑a∈Vh

‖ψa(∇×Ah)− hah‖2ωa

Demkowicz, Gopalakrishnan, & Schöberl (2009, 2012); Chaumont-Frelet & V.

(2021))

‖ψa(∇×Ah)−hah‖ωa≤Cst sup

v∈H∗(curl,ωa)‖∇×v‖ωa =1

〈R(Ah), ψav〉

≤CstCcont,PFW‖∇×(A− Ah)‖ωa

≤CL 61/2Ccont,PF

∑e∈Eh

‖∇×Ah − heh‖2ωe

Demkowicz, Gopalakrishnan, & Schöberl (2009); Ern, Chaumont-Frelet, & V.

(2021))

‖∇×Ah − heh‖ωe ≤ Cst sup

v∈H0(curl,ωe)‖∇×v‖ωe =1

〈R(Ah),v〉

≤ Cst‖∇×(A− Ah)‖ωe

∑a∈Vh

(2021))

〈R(Ah), ψav〉

≤CL 61/2Ccont,PF

∑e∈Eh

(2021))

〈R(Ah),v〉

∑a∈Vh

(2021))

〈R(Ah), ψav〉

≤CL 61/2Ccont,PF

∑e∈Eh

(2021))

〈R(Ah),v〉

∑a∈Vh

(2021))

〈R(Ah), ψav〉

≤CL 61/2Ccont,PF

∑e∈Eh

(2021))

〈R(Ah),v〉

∑a∈Vh

(2021))

〈R(Ah), ψav〉

≤CL 61/2Ccont,PF

∑e∈Eh

(2021))

〈R(Ah),v〉

Patchwise/broken patchwise flux equilibration

Patchwise flux equilibrationglobally equilibrated fluxPrager–Synge constant-free upperboundequilibration in several stages, moreexpensiveadditional layer for efficiencyp-robust

Broken patchwise flux equilibrationlocally equilibrated flux61/2, Ccont,PF, and CL in the upper bound;CL = 1 if Ω is convex and no mixed BCsequilibration in a single stage, cheaper,explicit for p = 0both estimator and efficiency on ωe

p-robust

Lift constant CL such that for all v ∈ H0,D(curl,Ω), there exists w ∈ H1(Ω) suchthat w ∈ H0,D(curl,Ω), ∇×w = ∇×v , and

‖∇w‖ ≤ CL‖∇×v‖

p-robust

‖∇w‖ ≤ CL‖∇×v‖

p-robust

‖∇w‖ ≤ CL‖∇×v‖

p-robust

‖∇w‖ ≤ CL‖∇×v‖

p-robust

‖∇w‖ ≤ CL‖∇×v‖

p-robust

‖∇w‖ ≤ CL‖∇×v‖

p-robust

‖∇w‖ ≤ CL‖∇×v‖

Outline

1 Introduction

3 The H(curl)-case

7 Conclusions

Stage 1: overconstrained Raviart–Thomas projection

Projection of ∇ψa×(∇×Ah) to a Raviart–Thomas space

For all vertices a ∈ Vh, consider p′ := minp,1-degree patchwise minimizations:

θah := arg min

vh∈RTp′ (Ta)∩H0(div,ωa)

∇·vh=−∇ψa·j(vh,rh)K =(∇ψa×(∇×Ah),rh)K ∀rh∈[P0(K )]3, ∀K∈Ta

‖∇ψa×(∇×Ah)− vh‖2ωa .

Comments∇ψa×(∇×Ah) 6∈RTp′(Ta) ∩ H0(div, ωa)additional orthogonality constraint

crucial for stage 2only possible thanks the lowest-order Galerkin orthogonality of Ahrequests minp,1

remainder δh :=∑

a∈Vhθa

hshould be zero (∼ partition of unity) but is notδh ∈RTp′(Th) ∩ H0,N(div,Ω) and ∇·δh = 0

θah := arg min

‖∇ψa×(∇×Ah)− vh‖2ωa .

remainder δh :=∑

a∈Vhθa

θah := arg min

‖∇ψa×(∇×Ah)− vh‖2ωa .

remainder δh :=∑

a∈Vhθa

θah := arg min

‖∇ψa×(∇×Ah)− vh‖2ωa .

remainder δh :=∑

a∈Vhθa

θah := arg min

‖∇ψa×(∇×Ah)− vh‖2ωa .

remainder δh :=∑

a∈Vhθa

Stage 2: divergence-free decomposition of the given divergence-freeRaviart-Thomas piecewise polynomial δh

Divergence-free decomposition of δh

For all tetrahedra K ∈ Th, consider (p + 1)-degree elementwise minimizations:

δah|K := arg min

vh∈RT1(K )∇·vh=0

vh·nK =IRT1 (ψaδh)·nK on ∂K

‖vh − IRT1 (ψaδh)‖2K ∀a ∈ VK when p = 0,

δah|K := arg min

vh∈RTp+1(K )∇·vh=0

vh·nK =ψaδh·nK on ∂K

‖vh − ψaδh‖2K ∀a ∈ VK when p ≥ 1.

Commentspatchwise contributions

δah ∈RTp+1(Ta) ∩ H0(div, ωa) and ∇·δa

h = 0 ∀a ∈ Vh

δah form a divergence-free decomposition of δh, δh =

∑a∈Vh

δah|K := arg min

h = 0 ∀a ∈ Vh

∑a∈Vh

δah|K := arg min

h = 0 ∀a ∈ Vh

∑a∈Vh

δah|K := arg min

h = 0 ∀a ∈ Vh

∑a∈Vh

Stage 2: divergence-free decomposition of the given divergence-freecurrent density j

Divergence-free decomposition of the current density jSet

jah := ψaj + θa

h − δah.

jah ∈RTp+1(Ta) ∩ H0(div, ωa),

∇·jah = 0,∑

a∈Vh

jah = j .

Stage 3: discrete patchwise equilibrated fluxes

Definition (Chaumont-Frelet, Vohralík (2021))For each vertex a ∈ Vh, solve the local constrained minimization problem

hah := arg min

vh∈Np+1(Ta)∩H0(curl,ωa)

∇×vh=jah

‖ψa(∇×Ah)− vh‖2ωa

and combinehh :=

∑a∈Vh

• a Ta

Key pointshomogeneous tangential BC on ∂ωa: hh ∈Np+1(Th) ∩ H(curl,Ω)

∇×hah =

∑a∈Vh

jah = j

hah := arg min

∇×vh=jah

and combinehh :=

∑a∈Vh

• a Ta

∇×hah =

∑a∈Vh

jah = j

hah := arg min

∇×vh=jah

and combinehh :=

∑a∈Vh

• a Ta

∇×hah =

∑a∈Vh

jah = j

hah := arg min

∇×vh=jah

and combinehh :=

∑a∈Vh

• a Ta

∇×hah =

∑a∈Vh

jah = j

hah := arg min

∇×vh=jah

and combinehh :=

∑a∈Vh

• a Ta

∇×hah =

∑a∈Vh

jah = j

Outline

1 Introduction

3 The H(curl)-case

7 Conclusions

H(curl) polynomial extension on a tetrahedron

Theorem (H(curl) polynomial extension on a tetrahedron Costabel & Mc-Intosh (2010); Demkowicz,

Gopalakrishnan, & Schöberl (2009); Chaumont-Frelet, Ern, & V. (2020))

Let ∅ ⊆ F ⊆ FK be a (sub)set of faces of a tetrahedron K . Then, for everypolynomial degree p ≥ 0, for all rK ∈RTp(K ) such that ∇·rK = 0, and for allrF ∈Nτ

p (ΓF ) such that rK ·nF = curlF (rF ) for all F ∈ F , there holds

minvp∈Np(K )∇×vp=rKvp|τF=rF

‖vp‖K ≤ Cst minv∈H(curl,K )∇×v=rKv |τF=rF

‖v‖K .

CommentsCst only depends on the shape-regularity of Kfor (pw) p-polynomial data rK , rF , minimization over Np(K ) is up to Cst asgood as minimization over the entire H(curl,K )extension to an edge patch: Chaumont-Frelet, Ern, & V. (2021)extension to a vertex patch: conjecture

H(curl) polynomial extension on a tetrahedron

Theorem (H(curl) polynomial extension on a tetrahedron Costabel & Mc-Intosh (2010); Demkowicz,

Gopalakrishnan, & Schöberl (2009); Chaumont-Frelet, Ern, & V. (2020))

Let ∅ ⊆ F ⊆ FK be a (sub)set of faces of a tetrahedron K . Then, for everypolynomial degree p ≥ 0, for all rK ∈RTp(K ) such that ∇·rK = 0, and for allrF ∈Nτ

p (ΓF ) such that rK ·nF = curlF (rF ) for all F ∈ F , there holds

minvp∈Np(K )∇×vp=rKvp|τF=rF

‖vp‖K ≤ Cst minv∈H(curl,K )∇×v=rKv |τF=rF

‖v‖K .

CommentsCst only depends on the shape-regularity of Kfor (pw) p-polynomial data rK , rF , minimization over Np(K ) is up to Cst asgood as minimization over the entire H(curl,K )extension to an edge patch: Chaumont-Frelet, Ern, & V. (2021)extension to a vertex patch: conjecture

Outline

1 Introduction

3 The H(curl)-case

7 Conclusions

Broken patchwise equilibration, smooth solution, h-refinement

1 1/2 1/4 1/8 1/16 1/32

10−4

10−2

‖∇×(A− Ah)‖

1 1/2 1/4 1/8 1/16 1/32

maxe∈Eh ηe/‖∇×(A− Ah)‖ωe

1 1/2 1/4 1/8 1/16 1/3225

η/‖∇×(A− Ah)‖

p = 0 p = 1 p = 2 p = 3

Broken patchwise equilibration, smooth solution, p-refinement

0 1 2 3 4 5 6 710−10

10−5

e−3p

‖∇×(A− Ah)‖

0 1 2 3 4 5 6 7

η/‖∇×(A− Ah)‖

N = 1 N = 2 N = 4 Unstrucutred

Broken patchwise equilibration, singular solution, adap. refinement

103 104 105

10−1

N2/3dofs

N1/3dofs

‖∇×(A− Ah)‖

103 104 105

ηcfree/‖∇×(A− Ah)‖

p = 0 p = 1 p = 2 p = 3

Broken patchwise equilibration, singular solution, adap. refinement

1.1e-3

2.2e-3

Estimated (left) and actual error (right), p = 3

Patchwise equilibration, H3 solution, h-refinement

10−1100

10−3

10−2

‖∇×(A− Ah)‖

10−11001.04

Effectivity index η/‖∇×(A− Ah)‖

error/eff. ind. estimatep = 1

error/eff. ind. estimatep = 2

Patchwise equilibration, H3 solution, p-refinement

1 2 3 4 5 6

10−4

10−3

10−2

‖∇×(A− Ah)‖

1 2 3 4 5 6

error estimatestructured mesh

error estimateunstructured mesh

Patchwise equilibration, singular solution, adap. refinement (p = 2)

104 105

10−1

N−2/3dof

‖∇×(A− Ah)‖

0 2 4 6 8 10

Adaptive refinement iter.

error estimate effectivity index

Patchwise equilibration, singular solution, adap. refinement (p = 2)

3.3e-3

6.6e-3

Estimated (left) and actual error (right), adaptive mesh refinement iteration #10.Top view (top) and side view (bottom)

Outline

1 Introduction

3 The H(curl)-case

7 Conclusions

ConclusionsConclusions

reliable, locally efficient, p-robust, and possibly constant-free estimatesdivergence-free decompositions of Raviart–Thomas piecewise polynomials

CHAUMONT-FRELET T., ERN A., VOHRALÍK M., Polynomial-degree-robustH(curl)-stability of discrete minimization in a tetrahedron, C. R. Math. Acad. Sci. Paris358 (2020), 1101–1110.

CHAUMONT-FRELET T., ERN A., VOHRALÍK M., Stable broken H(curl) polynomialextensions and p-robust a posteriori error estimates by broken patchwise equilibrationfor the curl–curl problem, Math. Comp. (2021), accepted for publication.

CHAUMONT-FRELET T., VOHRALÍK M., p-robust equilibrated flux reconstruction inH(curl) based on local minimizations. Application to a posteriori analysis of thecurl–curl problem, HAL Preprint 03227570, submitted for publication.

Thank you for your attention!M. Vohralík p-robust a posteriori estimation for the curl-curl problem 29 / 29

CHAUMONT-FRELET T., ERN A., VOHRALÍK M., Polynomial-degree-robustH(curl)-stability of discrete minimization in a tetrahedron, C. R. Math. Acad. Sci. Paris358 (2020), 1101–1110.

CHAUMONT-FRELET T., ERN A., VOHRALÍK M., Stable broken H(curl) polynomialextensions and p-robust a posteriori error estimates by broken patchwise equilibrationfor the curl–curl problem, Math. Comp. (2021), accepted for publication.

CHAUMONT-FRELET T., VOHRALÍK M., p-robust equilibrated flux reconstruction inH(curl) based on local minimizations. Application to a posteriori analysis of thecurl–curl problem, HAL Preprint 03227570, submitted for publication.

Thank you for your attention!M. Vohralík p-robust a posteriori estimation for the curl-curl problem 29 / 29

Théophile Chaumont-Frelet, Alexandre Ern, Martin Vohralík

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