Introduction to finite element exterior calculus

104
Introduction to finite element exterior calculus Ragnar Winther CMA, University of Oslo Norway

Transcript of Introduction to finite element exterior calculus

Page 1: Introduction to finite element exterior calculus

Introduction to finite element exterior calculus

Ragnar Winther

CMA, University of OsloNorway

Page 2: Introduction to finite element exterior calculus

Why finite element exterior calculus?

Recall the de Rham complex on the form:

R →H1(Ω)grad−−→ H(curl,Ω)

curl−−→ H(div,Ω)div−−→ L2(Ω) −→ 0

and the corresponding elasticity complex

T →H1(Ω; R3)ε−→ H(J,Ω; S)

J−→ H(div,Ω; S)div−−→ L2(Ω; R3) −→ 0.

Here J is the second order operator

Jτ = curl(curl τ)T

mapping symmetric matrix fields into symmetric matrix fields, andT is the six dimensional space of rigid motions.Reference: Arnold, Falk, Winther, Mixed finite elements for linearelasticity with weakly imposed symmetry, Math.Comp. 2007.

Page 3: Introduction to finite element exterior calculus

Why finite element exterior calculus?

Recall the de Rham complex on the form:

R →H1(Ω)grad−−→ H(curl,Ω)

curl−−→ H(div,Ω)div−−→ L2(Ω) −→ 0

and the corresponding elasticity complex

T →H1(Ω; R3)ε−→ H(J,Ω; S)

J−→ H(div,Ω; S)div−−→ L2(Ω; R3) −→ 0.

Here J is the second order operator

Jτ = curl(curl τ)T

mapping symmetric matrix fields into symmetric matrix fields, andT is the six dimensional space of rigid motions.

Reference: Arnold, Falk, Winther, Mixed finite elements for linearelasticity with weakly imposed symmetry, Math.Comp. 2007.

Page 4: Introduction to finite element exterior calculus

Why finite element exterior calculus?

Recall the de Rham complex on the form:

R →H1(Ω)grad−−→ H(curl,Ω)

curl−−→ H(div,Ω)div−−→ L2(Ω) −→ 0

and the corresponding elasticity complex

T →H1(Ω; R3)ε−→ H(J,Ω; S)

J−→ H(div,Ω; S)div−−→ L2(Ω; R3) −→ 0.

Here J is the second order operator

Jτ = curl(curl τ)T

mapping symmetric matrix fields into symmetric matrix fields, andT is the six dimensional space of rigid motions.Reference: Arnold, Falk, Winther, Mixed finite elements for linearelasticity with weakly imposed symmetry, Math.Comp. 2007.

Page 5: Introduction to finite element exterior calculus

Mapping of the domain

Consider a map φ : R3 → R3 mapping a domain Ω to Ω′.

Ω’Ωϕ

Desired commuting diagram:

R →H1(Ω′)grad−−→ H(curl,Ω′)

curl−−→ H(div,Ω′)div−−→ L2(Ω′) −→ 0yF 1

φ

yF cφ

yF dφ

yF 0φ

R → H1(Ω)grad−−→ H(curl,Ω)

curl−−→ H(div,Ω)div−−→ L2(Ω) −→ 0

Page 6: Introduction to finite element exterior calculus

Mapping of the domain

Consider a map φ : R3 → R3 mapping a domain Ω to Ω′.

Ω’Ωϕ

Desired commuting diagram:

R →H1(Ω′)grad−−→ H(curl,Ω′)

curl−−→ H(div,Ω′)div−−→ L2(Ω′) −→ 0yF 1

φ

yF cφ

yF dφ

yF 0φ

R → H1(Ω)grad−−→ H(curl,Ω)

curl−−→ H(div,Ω)div−−→ L2(Ω) −→ 0

Page 7: Introduction to finite element exterior calculus

Piola transforms

The proper “Piola transforms” are given by

(F 1φu)(x) = u(φ(x))

(F cφu)(x) = (Dφ(x))T u(φ(x)

(F dφ u)(x) = (det Dφ(x))(Dφ(x))−1u(φ(x))

(F 0φu)(x) = (det Dφ(x))u(φ(x))

Page 8: Introduction to finite element exterior calculus

Alternative de Rham complex

We replace the de Rham complex

R →H1(Ω)grad−−→ H(curl,Ω)

curl−−→ H(div,Ω)div−−→ L2(Ω) −→ 0

by an equivalent complex

R →HΛ0(Ω)d−→ HΛ1(Ω)

d−→ HΛ2(Ω)d−→ L2Λ3(Ω) −→ 0

In fact, the second complex can be defined for any dimension n.

Page 9: Introduction to finite element exterior calculus

Exterior calculus and differential forms

Instead of consider scalar fields and vector fields, i.e., functions ofthe form

u : Ω→ R or u : Ω→ R3

where Ω ⊂ R3, we consider instead functions of the form

u : Ω→ AltkRn

Here Ω is allowed to be a subset of Rn.

For any vector space V , the space AltkV is the space ofalternating k-linear maps on V

Page 10: Introduction to finite element exterior calculus

Exterior calculus and differential forms

Instead of consider scalar fields and vector fields, i.e., functions ofthe form

u : Ω→ R or u : Ω→ R3

where Ω ⊂ R3, we consider instead functions of the form

u : Ω→ AltkRn

Here Ω is allowed to be a subset of Rn.

For any vector space V , the space AltkV is the space ofalternating k-linear maps on V

Page 11: Introduction to finite element exterior calculus

The space AltkV

Assume dim V = n and that v1, v2, . . . vn is a basis. If ω ∈ AltkVand j1, j2, . . . , jk are integers in 1, 2, . . . , n thenω(vj1 , vj2 , . . . , vjk ) ∈ R.

Furthermore, ω is linear in each argument, and if two neighboringvectors change position then the value of ω change sign.Therefore, it is enough to specify ω(vj1 , vj2 , . . . , vjk ) for anincreasing sequence ji.As a consequence AltkV is a vector space with

dim AltkV =

(n

k

).

Note that dim AltkV = dim Altn−kV . The canonical map betweenthese spaces is the Hodge star operator.

Page 12: Introduction to finite element exterior calculus

The space AltkV

Assume dim V = n and that v1, v2, . . . vn is a basis. If ω ∈ AltkVand j1, j2, . . . , jk are integers in 1, 2, . . . , n thenω(vj1 , vj2 , . . . , vjk ) ∈ R.

Furthermore, ω is linear in each argument, and if two neighboringvectors change position then the value of ω change sign.Therefore, it is enough to specify ω(vj1 , vj2 , . . . , vjk ) for anincreasing sequence ji.

As a consequence AltkV is a vector space with

dim AltkV =

(n

k

).

Note that dim AltkV = dim Altn−kV . The canonical map betweenthese spaces is the Hodge star operator.

Page 13: Introduction to finite element exterior calculus

The space AltkV

Assume dim V = n and that v1, v2, . . . vn is a basis. If ω ∈ AltkVand j1, j2, . . . , jk are integers in 1, 2, . . . , n thenω(vj1 , vj2 , . . . , vjk ) ∈ R.

Furthermore, ω is linear in each argument, and if two neighboringvectors change position then the value of ω change sign.Therefore, it is enough to specify ω(vj1 , vj2 , . . . , vjk ) for anincreasing sequence ji.As a consequence AltkV is a vector space with

dim AltkV =

(n

k

).

Note that dim AltkV = dim Altn−kV . The canonical map betweenthese spaces is the Hodge star operator.

Page 14: Introduction to finite element exterior calculus

The space AltkV

Assume dim V = n and that v1, v2, . . . vn is a basis. If ω ∈ AltkVand j1, j2, . . . , jk are integers in 1, 2, . . . , n thenω(vj1 , vj2 , . . . , vjk ) ∈ R.

Furthermore, ω is linear in each argument, and if two neighboringvectors change position then the value of ω change sign.Therefore, it is enough to specify ω(vj1 , vj2 , . . . , vjk ) for anincreasing sequence ji.As a consequence AltkV is a vector space with

dim AltkV =

(n

k

).

Note that dim AltkV = dim Altn−kV . The canonical map betweenthese spaces is the Hodge star operator.

Page 15: Introduction to finite element exterior calculus

Interior and exterior products

If ω ∈ AltkV and v ∈ V then the interior product, or thecontraction of ω by v , ωyv ∈ Altk−1V is given by

ωyv(v1, v2, . . . vk−1) = ω(v , v1, v2, . . . vk−1)

If ω ∈ AltjV and η ∈ AltkV then the exterior product, or wedge

product, ω ∧ η ∈ Altj+k is given by

(ω ∧ η)(v1, . . . , vj+k)

=∑σ

(signσ)ω(vσ(1), . . . , vσ(j))η(vσ(j+1), . . . , vσ(j+k)), vi ∈ V ,

where the sum is over all permutations σ of 1, 2, . . . , j + k suchthat σ(1) < ... < σ(j) and σ(j + 1) < ... < σ(j + k).

Page 16: Introduction to finite element exterior calculus

Interior and exterior products

If ω ∈ AltkV and v ∈ V then the interior product, or thecontraction of ω by v , ωyv ∈ Altk−1V is given by

ωyv(v1, v2, . . . vk−1) = ω(v , v1, v2, . . . vk−1)

If ω ∈ AltjV and η ∈ AltkV then the exterior product, or wedge

product, ω ∧ η ∈ Altj+k is given by

(ω ∧ η)(v1, . . . , vj+k)

=∑σ

(signσ)ω(vσ(1), . . . , vσ(j))η(vσ(j+1), . . . , vσ(j+k)), vi ∈ V ,

where the sum is over all permutations σ of 1, 2, . . . , j + k suchthat σ(1) < ... < σ(j) and σ(j + 1) < ... < σ(j + k).

Page 17: Introduction to finite element exterior calculus

Basis for AltkRn

We let dxi be the natural dual basis in Alt1Rn given by

dxi (ej) = δi ,j

A basis for Alt2Rn is given by

dxi ∧ dxj , i < j ,

while

dxj1 ∧ dxj2 ∧ · · · ∧ dxjk , where ji is increasing

is a basis for AltkRn.

Page 18: Introduction to finite element exterior calculus

Basis for AltkRn

We let dxi be the natural dual basis in Alt1Rn given by

dxi (ej) = δi ,j

A basis for Alt2Rn is given by

dxi ∧ dxj , i < j ,

while

dxj1 ∧ dxj2 ∧ · · · ∧ dxjk , where ji is increasing

is a basis for AltkRn.

Page 19: Introduction to finite element exterior calculus

Basis for AltkRn

We let dxi be the natural dual basis in Alt1Rn given by

dxi (ej) = δi ,j

A basis for Alt2Rn is given by

dxi ∧ dxj , i < j ,

while

dxj1 ∧ dxj2 ∧ · · · ∧ dxjk , where ji is increasing

is a basis for AltkRn.

Page 20: Introduction to finite element exterior calculus

Any ω ∈ AltkRn can be represented uniquely on the form∑σ

uσdxσ(1) ∧ dxσ(2) ∧ . . . ∧ dxσ(k)

where the sum is over all incresing maps σ mapping 1, 2, . . . , kinto 1, 2, . . . , n, and uσ ∈ R.

Furthermore,

dxσ(1) ∧ dxσ(2) ∧ . . . ∧ dxσ(k)(v1, . . . vk) = det(dxσ(i)vj)

Page 21: Introduction to finite element exterior calculus

Any ω ∈ AltkRn can be represented uniquely on the form∑σ

uσdxσ(1) ∧ dxσ(2) ∧ . . . ∧ dxσ(k)

where the sum is over all incresing maps σ mapping 1, 2, . . . , kinto 1, 2, . . . , n, and uσ ∈ R.

Furthermore,

dxσ(1) ∧ dxσ(2) ∧ . . . ∧ dxσ(k)(v1, . . . vk) = det(dxσ(i)vj)

Page 22: Introduction to finite element exterior calculus

Correspondences

Altk and vector fields:

Alt0 R3 = R c ↔ c

Alt1 R3 ∼=−→ R3 u1 dx1 + u2 dx2 + u3 dx3 ↔ u

Alt2 R3 ∼=−→ R3 u3 dx1 ∧ dx2 − u2 dx1 ∧ dx3 + u1 dx2 ∧ dx3 ↔ u

Alt3 R3 ∼=−→ R c ↔ c dx1 ∧ dx2 ∧ dx3

Exterior product:

∧ : Alt1 R3 × Alt1 R3 → Alt2 R3 × : R3 × R3 → R3

∧ : Alt1 R3 × Alt2 R3 → Alt3 R3 · : R3 × R3 → R

Page 23: Introduction to finite element exterior calculus

Correspondences

Altk and vector fields:

Alt0 R3 = R c ↔ c

Alt1 R3 ∼=−→ R3 u1 dx1 + u2 dx2 + u3 dx3 ↔ u

Alt2 R3 ∼=−→ R3 u3 dx1 ∧ dx2 − u2 dx1 ∧ dx3 + u1 dx2 ∧ dx3 ↔ u

Alt3 R3 ∼=−→ R c ↔ c dx1 ∧ dx2 ∧ dx3

Exterior product:

∧ : Alt1 R3 × Alt1 R3 → Alt2 R3 × : R3 × R3 → R3

∧ : Alt1 R3 × Alt2 R3 → Alt3 R3 · : R3 × R3 → R

Page 24: Introduction to finite element exterior calculus

Differential forms

Differential forms are functions of the form

ω : Ω→ Altk ≡ Altk Rn.

We let Λk = Λk(Ω) = C∞(Ω; Altk). For such smooth differentiableforms the exterior derivative, d : Λk → Λk+1 is defined by

dωx(v1, v2, . . . , vk+1) =k+1∑j=1

(−1)j+1∂vjωx(v1, . . . , vj , . . . , vk+1),

ω ∈ Λk , v1, . . . , vk+1 ∈ Rn

Using the fact that ∂vi∂vj = ∂vj∂vi it follows that d d = 0.

Page 25: Introduction to finite element exterior calculus

Differential forms

Differential forms are functions of the form

ω : Ω→ Altk ≡ Altk Rn.

We let Λk = Λk(Ω) = C∞(Ω; Altk). For such smooth differentiableforms the exterior derivative, d : Λk → Λk+1 is defined by

dωx(v1, v2, . . . , vk+1) =k+1∑j=1

(−1)j+1∂vjωx(v1, . . . , vj , . . . , vk+1),

ω ∈ Λk , v1, . . . , vk+1 ∈ Rn

Using the fact that ∂vi∂vj = ∂vj∂vi it follows that d d = 0.

Page 26: Introduction to finite element exterior calculus

Differential forms

Differential forms are functions of the form

ω : Ω→ Altk ≡ Altk Rn.

We let Λk = Λk(Ω) = C∞(Ω; Altk). For such smooth differentiableforms the exterior derivative, d : Λk → Λk+1 is defined by

dωx(v1, v2, . . . , vk+1) =k+1∑j=1

(−1)j+1∂vjωx(v1, . . . , vj , . . . , vk+1),

ω ∈ Λk , v1, . . . , vk+1 ∈ Rn

Using the fact that ∂vi∂vj = ∂vj∂vi it follows that d d = 0.

Page 27: Introduction to finite element exterior calculus

Check of d d = 0

dωx(v1, v2, . . . , vk+1) =k+1∑j=1

(−1)j+1∂vjωx(v1, . . . , vj , . . . , vk+1),

ω ∈ Λk , v1, . . . , vk+1 ∈ Rn

Start with a zero form ω.

dωx(v) = ∂vωx

(d d)ω(v1, v2) = ∂v1dω(v2)− ∂v2dω(v1) = 0.

Page 28: Introduction to finite element exterior calculus

Check of d d = 0

dωx(v1, v2, . . . , vk+1) =k+1∑j=1

(−1)j+1∂vjωx(v1, . . . , vj , . . . , vk+1),

ω ∈ Λk , v1, . . . , vk+1 ∈ Rn

Start with a zero form ω.

dωx(v) = ∂vωx

(d d)ω(v1, v2) = ∂v1dω(v2)− ∂v2dω(v1) = 0.

Page 29: Introduction to finite element exterior calculus

Check of d d = 0

dωx(v1, v2, . . . , vk+1) =k+1∑j=1

(−1)j+1∂vjωx(v1, . . . , vj , . . . , vk+1),

ω ∈ Λk , v1, . . . , vk+1 ∈ Rn

Start with a zero form ω.

dωx(v) = ∂vωx

(d d)ω(v1, v2) = ∂v1dω(v2)− ∂v2dω(v1) = 0.

Page 30: Introduction to finite element exterior calculus

Representation by a basis

If ω is given on the form

ωx =∑σ

uσdxσ(1) ∧ dxσ(2) ∧ . . . ∧ dxσ(k),

where each uσ is a scalar field on Ω then

dωx =∑σ

n∑j=1

∂uσ∂xj

dxj ∧ dxσ(1) ∧ dxσ(2) ∧ . . . ∧ dxσ(k)

Page 31: Introduction to finite element exterior calculus

The de Rham complex and differential forms

Using differential forms the de Rham complex (smooth version)can be written as

R →Λ0 d−→ Λ1 d−→ · · · d−→ Λn −→ 0.

Page 32: Introduction to finite element exterior calculus

Mapping of the domain

Ω’Ωϕ

The corresponding pull backs, φ∗, which maps Λk(Ω′) to Λk(Ω), isgiven by

(φ∗ω)x(v1, v2, . . . , vk) = ωφ(x)(Dφx(v1),Dφx(v2), . . . ,Dφx(vk)),

where Dφx is the derivative of φ at x mapping TxΩ into Tφ(x)Ω′.This expression should be compared with the expressions we hadfor the Piola maps before.

Page 33: Introduction to finite element exterior calculus

Mapping of the domain

Ω’Ωϕ

The corresponding pull backs, φ∗, which maps Λk(Ω′) to Λk(Ω), isgiven by

(φ∗ω)x(v1, v2, . . . , vk) = ωφ(x)(Dφx(v1),Dφx(v2), . . . ,Dφx(vk)),

where Dφx is the derivative of φ at x mapping TxΩ into Tφ(x)Ω′.

This expression should be compared with the expressions we hadfor the Piola maps before.

Page 34: Introduction to finite element exterior calculus

Mapping of the domain

Ω’Ωϕ

The corresponding pull backs, φ∗, which maps Λk(Ω′) to Λk(Ω), isgiven by

(φ∗ω)x(v1, v2, . . . , vk) = ωφ(x)(Dφx(v1),Dφx(v2), . . . ,Dφx(vk)),

where Dφx is the derivative of φ at x mapping TxΩ into Tφ(x)Ω′.This expression should be compared with the expressions we hadfor the Piola maps before.

Page 35: Introduction to finite element exterior calculus

Commuting diagram

R →Λ0(Ω′)d−→ Λ1(Ω′)

d−→ · · · d−→ Λn(Ω′) −→ 0.yφ∗ yφ∗ yφ∗R → Λ0(Ω)

d−→ Λ1(Ω)d−→ · · · d−→ Λn(Ω) −→ 0.

So the pullback commutes with the exterior derivative, i.e.,

φ∗(dω) = d(φ∗ω), ω ∈ Λk(Ω′),

and distributes with respect to the wedge product:

φ∗(ω ∧ η) = φ∗ω ∧ φ∗η.

Note this also gives us a way to define the exterior derivative on amanifold.Key identity: (ψ φ)∗ = φ∗ ψ∗

Page 36: Introduction to finite element exterior calculus

Commuting diagram

R →Λ0(Ω′)d−→ Λ1(Ω′)

d−→ · · · d−→ Λn(Ω′) −→ 0.yφ∗ yφ∗ yφ∗R → Λ0(Ω)

d−→ Λ1(Ω)d−→ · · · d−→ Λn(Ω) −→ 0.

So the pullback commutes with the exterior derivative, i.e.,

φ∗(dω) = d(φ∗ω), ω ∈ Λk(Ω′),

and distributes with respect to the wedge product:

φ∗(ω ∧ η) = φ∗ω ∧ φ∗η.

Note this also gives us a way to define the exterior derivative on amanifold.Key identity: (ψ φ)∗ = φ∗ ψ∗

Page 37: Introduction to finite element exterior calculus

Commuting diagram

R →Λ0(Ω′)d−→ Λ1(Ω′)

d−→ · · · d−→ Λn(Ω′) −→ 0.yφ∗ yφ∗ yφ∗R → Λ0(Ω)

d−→ Λ1(Ω)d−→ · · · d−→ Λn(Ω) −→ 0.

So the pullback commutes with the exterior derivative, i.e.,

φ∗(dω) = d(φ∗ω), ω ∈ Λk(Ω′),

and distributes with respect to the wedge product:

φ∗(ω ∧ η) = φ∗ω ∧ φ∗η.

Note this also gives us a way to define the exterior derivative on amanifold.

Key identity: (ψ φ)∗ = φ∗ ψ∗

Page 38: Introduction to finite element exterior calculus

Commuting diagram

R →Λ0(Ω′)d−→ Λ1(Ω′)

d−→ · · · d−→ Λn(Ω′) −→ 0.yφ∗ yφ∗ yφ∗R → Λ0(Ω)

d−→ Λ1(Ω)d−→ · · · d−→ Λn(Ω) −→ 0.

So the pullback commutes with the exterior derivative, i.e.,

φ∗(dω) = d(φ∗ω), ω ∈ Λk(Ω′),

and distributes with respect to the wedge product:

φ∗(ω ∧ η) = φ∗ω ∧ φ∗η.

Note this also gives us a way to define the exterior derivative on amanifold.Key identity: (ψ φ)∗ = φ∗ ψ∗

Page 39: Introduction to finite element exterior calculus

Exterior derivative and wedge product

By combining the wedge product with the exterior derivative weobtain a Leibniz rule of the form

d(ω ∧ η) = dω ∧ η + (−1)jω ∧ dη, ω ∈ Λj , η ∈ Λk .

The pullback of the inclusion ∂Ω → Ω is the trace map, Tr. Thevanishing of Trω for ω ∈ Λk(Ω) does not imply that ωx vanishesfor x ∈ ∂Ω, only that it vanishes when applied to k-tuples oftangent vectors to ∂Ω.

Since Tr is a pull back we will have

Tr d = d∂Ω Tr

Page 40: Introduction to finite element exterior calculus

Exterior derivative and wedge product

By combining the wedge product with the exterior derivative weobtain a Leibniz rule of the form

d(ω ∧ η) = dω ∧ η + (−1)jω ∧ dη, ω ∈ Λj , η ∈ Λk .

The pullback of the inclusion ∂Ω → Ω is the trace map, Tr. Thevanishing of Trω for ω ∈ Λk(Ω) does not imply that ωx vanishesfor x ∈ ∂Ω, only that it vanishes when applied to k-tuples oftangent vectors to ∂Ω.

Since Tr is a pull back we will have

Tr d = d∂Ω Tr

Page 41: Introduction to finite element exterior calculus

Exterior derivative and wedge product

By combining the wedge product with the exterior derivative weobtain a Leibniz rule of the form

d(ω ∧ η) = dω ∧ η + (−1)jω ∧ dη, ω ∈ Λj , η ∈ Λk .

The pullback of the inclusion ∂Ω → Ω is the trace map, Tr. Thevanishing of Trω for ω ∈ Λk(Ω) does not imply that ωx vanishesfor x ∈ ∂Ω, only that it vanishes when applied to k-tuples oftangent vectors to ∂Ω.

Since Tr is a pull back we will have

Tr d = d∂Ω Tr

Page 42: Introduction to finite element exterior calculus

Integration of differential forms

If ω ∈ Λn(Ω), Ω ⊂ Rn, with the representation

ωx = uxdx1 ∧ dx2 ∧ . . . ∧ dxn,

then we can define ∫Ωω =

∫Ω

u dx

If φ : Ω→ Ω′ then we have∫Ωφ∗ω =

∫Ω′ω

Note that this gives us a way to define the integral of an n-form ona manifold.

Page 43: Introduction to finite element exterior calculus

Integration of differential forms

If ω ∈ Λn(Ω), Ω ⊂ Rn, with the representation

ωx = uxdx1 ∧ dx2 ∧ . . . ∧ dxn,

then we can define ∫Ωω =

∫Ω

u dx

If φ : Ω→ Ω′ then we have∫Ωφ∗ω =

∫Ω′ω

Note that this gives us a way to define the integral of an n-form ona manifold.

Page 44: Introduction to finite element exterior calculus

Integration of differential forms

If ω ∈ Λn(Ω), Ω ⊂ Rn, with the representation

ωx = uxdx1 ∧ dx2 ∧ . . . ∧ dxn,

then we can define ∫Ωω =

∫Ω

u dx

If φ : Ω→ Ω′ then we have∫Ωφ∗ω =

∫Ω′ω

Note that this gives us a way to define the integral of an n-form ona manifold.

Page 45: Introduction to finite element exterior calculus

Stokes theorem

With the three concepts exterior derivative, trace operator, andintegral we can formulate Stokes theorem as∫

Ωdω =

∫∂Ω

Trω, ω ∈ Λn−1(Ω)

Furthermore, by combining this with the Leibniz rule we obtain thefollowing integration by parts identity∫

Ωdω ∧ η = (−1)k−1

∫Ωω ∧ dη +

∫∂Ω

Trω ∧ Tr η

where ω ∈ Λk , η ∈ Λn−k−1.

Page 46: Introduction to finite element exterior calculus

Stokes theorem

With the three concepts exterior derivative, trace operator, andintegral we can formulate Stokes theorem as∫

Ωdω =

∫∂Ω

Trω, ω ∈ Λn−1(Ω)

Furthermore, by combining this with the Leibniz rule we obtain thefollowing integration by parts identity∫

Ωdω ∧ η = (−1)k−1

∫Ωω ∧ dη +

∫∂Ω

Trω ∧ Tr η

where ω ∈ Λk , η ∈ Λn−k−1.

Page 47: Introduction to finite element exterior calculus

L2 differential forms

If ω is a k form given by

ω =∑σ

uσdxσ(1) ∧ dxσ(2) ∧ . . . ∧ dxσ(k)

then ω is defined to be in ∈ L2Λk(Ω) if each of the coefficients uσare in L2.

Furthernmore, we let

HΛk(Ω) = ω ∈ L2Λk(Ω) | dω ∈ L2Λk+1(Ω)

We then ontain the L2 de Rham complex:

0→ HΛ0(Ω)d−→ HΛ1(Ω)

d−→ · · · d−→ HΛn(Ω)→ 0

Page 48: Introduction to finite element exterior calculus

L2 differential forms

If ω is a k form given by

ω =∑σ

uσdxσ(1) ∧ dxσ(2) ∧ . . . ∧ dxσ(k)

then ω is defined to be in ∈ L2Λk(Ω) if each of the coefficients uσare in L2.

Furthernmore, we let

HΛk(Ω) = ω ∈ L2Λk(Ω) | dω ∈ L2Λk+1(Ω)

We then ontain the L2 de Rham complex:

0→ HΛ0(Ω)d−→ HΛ1(Ω)

d−→ · · · d−→ HΛn(Ω)→ 0

Page 49: Introduction to finite element exterior calculus

L2 differential forms

If ω is a k form given by

ω =∑σ

uσdxσ(1) ∧ dxσ(2) ∧ . . . ∧ dxσ(k)

then ω is defined to be in ∈ L2Λk(Ω) if each of the coefficients uσare in L2.

Furthernmore, we let

HΛk(Ω) = ω ∈ L2Λk(Ω) | dω ∈ L2Λk+1(Ω)

We then ontain the L2 de Rham complex:

0→ HΛ0(Ω)d−→ HΛ1(Ω)

d−→ · · · d−→ HΛn(Ω)→ 0

Page 50: Introduction to finite element exterior calculus

Polynomial de Rham complex

We define

Pr Λk = ω ∈ Λk |ω(v1, . . . vk) ∈ Pr , ∀v1, . . . vk

Alternatively, Pr Λk consists of all∑σ

uσdxσ(1) ∧ dxσ(2) ∧ . . . ∧ dxσ(k)

where the coefficients uσ are in Pr .The polynomial de Rham complex

0→ Pr Λ0 d−→ Pr−1Λ1 d−→ · · · d−→ Pr−nΛn → 0

is exact.

Page 51: Introduction to finite element exterior calculus

Polynomial de Rham complex

We define

Pr Λk = ω ∈ Λk |ω(v1, . . . vk) ∈ Pr , ∀v1, . . . vk

Alternatively, Pr Λk consists of all∑σ

uσdxσ(1) ∧ dxσ(2) ∧ . . . ∧ dxσ(k)

where the coefficients uσ are in Pr .

The polynomial de Rham complex

0→ Pr Λ0 d−→ Pr−1Λ1 d−→ · · · d−→ Pr−nΛn → 0

is exact.

Page 52: Introduction to finite element exterior calculus

Polynomial de Rham complex

We define

Pr Λk = ω ∈ Λk |ω(v1, . . . vk) ∈ Pr , ∀v1, . . . vk

Alternatively, Pr Λk consists of all∑σ

uσdxσ(1) ∧ dxσ(2) ∧ . . . ∧ dxσ(k)

where the coefficients uσ are in Pr .The polynomial de Rham complex

0→ Pr Λ0 d−→ Pr−1Λ1 d−→ · · · d−→ Pr−nΛn → 0

is exact.

Page 53: Introduction to finite element exterior calculus

The Kozul complex

The Koszul differential κ of a k-form ω is the (k − 1)-form given by

(κω)x(v1, . . . , vk−1) = ωxyx(v1, . . . , vk−1)

= ωx

(x , v1, . . . , vk−1

),

For each r , κ maps Pr−1Λk to Pr Λk−1, and the Koszul complex

0→ Pr−nΛn κ−→ Pr−n+1Λn−1 κ−→ · · · κ−→ Pr Λ0 −→ R→ 0,

is exact.

Page 54: Introduction to finite element exterior calculus

The Kozul complex

The Koszul differential κ of a k-form ω is the (k − 1)-form given by

(κω)x(v1, . . . , vk−1) = ωxyx(v1, . . . , vk−1)

= ωx

(x , v1, . . . , vk−1

),

For each r , κ maps Pr−1Λk to Pr Λk−1,

and the Koszul complex

0→ Pr−nΛn κ−→ Pr−n+1Λn−1 κ−→ · · · κ−→ Pr Λ0 −→ R→ 0,

is exact.

Page 55: Introduction to finite element exterior calculus

The Kozul complex

The Koszul differential κ of a k-form ω is the (k − 1)-form given by

(κω)x(v1, . . . , vk−1) = ωxyx(v1, . . . , vk−1)

= ωx

(x , v1, . . . , vk−1

),

For each r , κ maps Pr−1Λk to Pr Λk−1, and the Koszul complex

0→ Pr−nΛn κ−→ Pr−n+1Λn−1 κ−→ · · · κ−→ Pr Λ0 −→ R→ 0,

is exact.

Page 56: Introduction to finite element exterior calculus

The spaces P−r Λk

The operators d and κ are related by the homotopy relation

(dκ+ κd)ω = (r + k)ω, ω ∈ Hr Λk ,

where Hr denotes the homogeneous polynomials of degree r .

As a consequence we obtain the identity

Pr Λk = Pr−1Λk + κHr−1Λk+1 + dHr+1Λk−1

We then define the space P−r Λk by

P−r Λk = Pr−1Λk + κPr−1Λk+1.

We note that P−r Λ0 = Pr Λ0 and P−r Λn = Pr−1Λn. Furthermore,

0→ P−r Λ0 d−→ P−r Λ1 d−→ · · · d−→ P−r Λn → 0

is an exact complex, and the space P−r Λk is affine invariant.

Page 57: Introduction to finite element exterior calculus

The spaces P−r Λk

The operators d and κ are related by the homotopy relation

(dκ+ κd)ω = (r + k)ω, ω ∈ Hr Λk ,

where Hr denotes the homogeneous polynomials of degree r .As a consequence we obtain the identity

Pr Λk = Pr−1Λk + κHr−1Λk+1 + dHr+1Λk−1

We then define the space P−r Λk by

P−r Λk = Pr−1Λk + κPr−1Λk+1.

We note that P−r Λ0 = Pr Λ0 and P−r Λn = Pr−1Λn. Furthermore,

0→ P−r Λ0 d−→ P−r Λ1 d−→ · · · d−→ P−r Λn → 0

is an exact complex, and the space P−r Λk is affine invariant.

Page 58: Introduction to finite element exterior calculus

The spaces P−r Λk

The operators d and κ are related by the homotopy relation

(dκ+ κd)ω = (r + k)ω, ω ∈ Hr Λk ,

where Hr denotes the homogeneous polynomials of degree r .As a consequence we obtain the identity

Pr Λk = Pr−1Λk + κHr−1Λk+1 + dHr+1Λk−1

We then define the space P−r Λk by

P−r Λk = Pr−1Λk + κPr−1Λk+1.

We note that P−r Λ0 = Pr Λ0 and P−r Λn = Pr−1Λn. Furthermore,

0→ P−r Λ0 d−→ P−r Λ1 d−→ · · · d−→ P−r Λn → 0

is an exact complex, and the space P−r Λk is affine invariant.

Page 59: Introduction to finite element exterior calculus

The spaces P−r Λk

The operators d and κ are related by the homotopy relation

(dκ+ κd)ω = (r + k)ω, ω ∈ Hr Λk ,

where Hr denotes the homogeneous polynomials of degree r .As a consequence we obtain the identity

Pr Λk = Pr−1Λk + κHr−1Λk+1 + dHr+1Λk−1

We then define the space P−r Λk by

P−r Λk = Pr−1Λk + κPr−1Λk+1.

We note that P−r Λ0 = Pr Λ0 and P−r Λn = Pr−1Λn. Furthermore,

0→ P−r Λ0 d−→ P−r Λ1 d−→ · · · d−→ P−r Λn → 0

is an exact complex, and the space P−r Λk is affine invariant.

Page 60: Introduction to finite element exterior calculus

Characterization of affine invariant polynomial spaces

TheoremAssume that the polynomial space QΛk , satisfiesPr−1Λk ( QΛk ( Pr Λk . Then

QΛk = P−r Λk or QΛk = Pr−1Λk + dPr+1Λk−1

Page 61: Introduction to finite element exterior calculus

The four exact sequences ending with Pr Λ3(T ) in 3D

0→ Pr+1Λ0 d−→ P−r+1Λ1 d−→ P−r+1Λ2 d−→ Pr Λ3 → 0

0→ Pr+2Λ0 d−→ Pr+1Λ1 d−→ P−r+1Λ2 d−→ Pr Λ3 → 0

0→ Pr+2Λ0 d−→ P−r+2Λ1 d−→ Pr+1Λ2 d−→ Pr Λ3 → 0

0→ Pr+3Λ0 d−→ Pr+2Λ1 d−→ Pr+1Λ2 d−→ Pr Λ3 → 0

In general there are 2n−1 exact sequences composed of the familiesof P and P− spaces.

Page 62: Introduction to finite element exterior calculus

The four exact sequences ending with Pr Λ3(T ) in 3D

0→ Pr+1Λ0 d−→ P−r+1Λ1 d−→ P−r+1Λ2 d−→ Pr Λ3 → 0

0→ Pr+2Λ0 d−→ Pr+1Λ1 d−→ P−r+1Λ2 d−→ Pr Λ3 → 0

0→ Pr+2Λ0 d−→ P−r+2Λ1 d−→ Pr+1Λ2 d−→ Pr Λ3 → 0

0→ Pr+3Λ0 d−→ Pr+2Λ1 d−→ Pr+1Λ2 d−→ Pr Λ3 → 0

In general there are 2n−1 exact sequences composed of the familiesof P and P− spaces.

Page 63: Introduction to finite element exterior calculus

The four sequences ending with P0Λ3(T ) in 3D

0→ grad−−→ curl−−→ div−−→ → 0

0→ grad−−→ curl−−→ div−−→ → 0

0→ grad−−→ curl−−→ div−−→ → 0

0→ grad−−→ curl−−→ div−−→ → 0

Page 64: Introduction to finite element exterior calculus

Piecewise smooth differential formsIt is a consequence of Stokes theorem that a piecewise smoothk–form ω, with respect to a simplicial mesh Th of Ω, is in HΛk(Ω)if and only if the trace of ω, Trω, is continuous on the interfaces.

Here Trω is defined by restricting thespatial variable x to the interface, andby applying ω only to tangent vectors ofthe interface.

Page 65: Introduction to finite element exterior calculus

Degrees of freedom

To obtain finite element differential forms—not just pw polynomials—we

need degrees of freedom, i.e., a decomposition of the dual spaces

(Pr Λk(T ))∗ and (P−r Λk(T ))∗ (with T a simplex), into subspaces

associated to subsimplices f of T .

DOF for Pr Λk(T ): to a subsimplex f of dimension d we associate

ω 7→∫

fTrf ω ∧ η, η ∈ P−r+k−dΛd−k(f )

DOF for P−r Λk(T ):

ω 7→∫

fTrf ω ∧ η, η ∈ Pr+k−d−1Λd−k(f )

Given a triangulation T , we can then define Pr Λk(T ), P−r Λk(T ). They

are subspaces of HΛk(Ω).

Page 66: Introduction to finite element exterior calculus

Degrees of freedom

To obtain finite element differential forms—not just pw polynomials—we

need degrees of freedom, i.e., a decomposition of the dual spaces

(Pr Λk(T ))∗ and (P−r Λk(T ))∗ (with T a simplex), into subspaces

associated to subsimplices f of T .

DOF for Pr Λk(T ): to a subsimplex f of dimension d we associate

ω 7→∫

fTrf ω ∧ η, η ∈ P−r+k−dΛd−k(f )

DOF for P−r Λk(T ):

ω 7→∫

fTrf ω ∧ η, η ∈ Pr+k−d−1Λd−k(f )

Given a triangulation T , we can then define Pr Λk(T ), P−r Λk(T ). They

are subspaces of HΛk(Ω).

Page 67: Introduction to finite element exterior calculus

Degrees of freedom

To obtain finite element differential forms—not just pw polynomials—we

need degrees of freedom, i.e., a decomposition of the dual spaces

(Pr Λk(T ))∗ and (P−r Λk(T ))∗ (with T a simplex), into subspaces

associated to subsimplices f of T .

DOF for Pr Λk(T ): to a subsimplex f of dimension d we associate

ω 7→∫

fTrf ω ∧ η, η ∈ P−r+k−dΛd−k(f )

DOF for P−r Λk(T ):

ω 7→∫

fTrf ω ∧ η, η ∈ Pr+k−d−1Λd−k(f )

Given a triangulation T , we can then define Pr Λk(T ), P−r Λk(T ). They

are subspaces of HΛk(Ω).

Page 68: Introduction to finite element exterior calculus

Degrees of freedom

To obtain finite element differential forms—not just pw polynomials—we

need degrees of freedom, i.e., a decomposition of the dual spaces

(Pr Λk(T ))∗ and (P−r Λk(T ))∗ (with T a simplex), into subspaces

associated to subsimplices f of T .

DOF for Pr Λk(T ): to a subsimplex f of dimension d we associate

ω 7→∫

fTrf ω ∧ η, η ∈ P−r+k−dΛd−k(f )

DOF for P−r Λk(T ):

ω 7→∫

fTrf ω ∧ η, η ∈ Pr+k−d−1Λd−k(f )

Given a triangulation T , we can then define Pr Λk(T ), P−r Λk(T ). They

are subspaces of HΛk(Ω).

Page 69: Introduction to finite element exterior calculus

De Rham cohomology

0 → Λ0(Ω)d−→ Λ1(Ω)

d−→ Λ2(Ω) → 0

0 → C∞(Ω)grad−−→ C∞(Ω; R2)

rot−→ C∞(Ω) → 0

Page 70: Introduction to finite element exterior calculus

Cohomology

The de Rham complex

HΛk−1(Ω)dk−1

−−−→ HΛk(Ω)dk

−→ HΛk+1(Ω)

is called exact if for all k,

Zk := ker dk = range dk−1 =: Bk .

In general, Bk ⊂ Zk and we assume throughout that the kthcohomology group Zk/Bk is finite dimensional.The space of harmonic k-forms, Hk , consists of all q ∈ Zk suchthat

〈q, µ〉 = 0 µ ∈ Bk .

This leads to the Hodge decompositionHΛk(Ω) = Zk ⊕ Zk⊥ = Bk ⊕ Hk ⊕ Zk⊥. Note that Hk ∼= Zk/Bk .

Page 71: Introduction to finite element exterior calculus

Cohomology

The de Rham complex

HΛk−1(Ω)dk−1

−−−→ HΛk(Ω)dk

−→ HΛk+1(Ω)

is called exact if for all k,

Zk := ker dk = range dk−1 =: Bk .

In general, Bk ⊂ Zk and we assume throughout that the kthcohomology group Zk/Bk is finite dimensional.

The space of harmonic k-forms, Hk , consists of all q ∈ Zk suchthat

〈q, µ〉 = 0 µ ∈ Bk .

This leads to the Hodge decompositionHΛk(Ω) = Zk ⊕ Zk⊥ = Bk ⊕ Hk ⊕ Zk⊥. Note that Hk ∼= Zk/Bk .

Page 72: Introduction to finite element exterior calculus

Cohomology

The de Rham complex

HΛk−1(Ω)dk−1

−−−→ HΛk(Ω)dk

−→ HΛk+1(Ω)

is called exact if for all k,

Zk := ker dk = range dk−1 =: Bk .

In general, Bk ⊂ Zk and we assume throughout that the kthcohomology group Zk/Bk is finite dimensional.The space of harmonic k-forms, Hk , consists of all q ∈ Zk suchthat

〈q, µ〉 = 0 µ ∈ Bk .

This leads to the Hodge decompositionHΛk(Ω) = Zk ⊕ Zk⊥ = Bk ⊕ Hk ⊕ Zk⊥. Note that Hk ∼= Zk/Bk .

Page 73: Introduction to finite element exterior calculus

Cohomology

The de Rham complex

HΛk−1(Ω)dk−1

−−−→ HΛk(Ω)dk

−→ HΛk+1(Ω)

is called exact if for all k,

Zk := ker dk = range dk−1 =: Bk .

In general, Bk ⊂ Zk and we assume throughout that the kthcohomology group Zk/Bk is finite dimensional.The space of harmonic k-forms, Hk , consists of all q ∈ Zk suchthat

〈q, µ〉 = 0 µ ∈ Bk .

This leads to the Hodge decompositionHΛk(Ω) = Zk ⊕ Zk⊥ = Bk ⊕ Hk ⊕ Zk⊥. Note that Hk ∼= Zk/Bk .

Page 74: Introduction to finite element exterior calculus

Hodge Laplace problem

Λk−1(Ω)dk−1

−−−→ Λk(Ω)dk

−→ Λk+1(Ω)

Formally: Given f ∈ Λk , find u ∈ Λk such that

(dk−1δk−1 + δkdk)u = f .

Here δk is a formal adjoint of dk .

The following mixed formulation is always well-posed: Givenf ∈ L2Λk(Ω), find σ ∈ HΛk−1, u ∈ HΛk and p ∈ Hk such that

〈σ, τ〉 − 〈dτ, u〉 = 0 ∀τ ∈ HΛk−1

〈dσ, v〉+ 〈du, dv〉+ 〈p, v〉 = 〈f , v〉 ∀v ∈ HΛk

〈u, q〉 = 0 ∀q ∈ Hk

Page 75: Introduction to finite element exterior calculus

Hodge Laplace problem

Λk−1(Ω)dk−1

−−−→ Λk(Ω)dk

−→ Λk+1(Ω)

Formally: Given f ∈ Λk , find u ∈ Λk such that

(dk−1δk−1 + δkdk)u = f .

Here δk is a formal adjoint of dk .

The following mixed formulation is always well-posed: Givenf ∈ L2Λk(Ω), find σ ∈ HΛk−1, u ∈ HΛk and p ∈ Hk such that

〈σ, τ〉 − 〈dτ, u〉 = 0 ∀τ ∈ HΛk−1

〈dσ, v〉+ 〈du, dv〉+ 〈p, v〉 = 〈f , v〉 ∀v ∈ HΛk

〈u, q〉 = 0 ∀q ∈ Hk

Page 76: Introduction to finite element exterior calculus

Hodge Laplace problem

Λk−1(Ω)dk−1

−−−→ Λk(Ω)dk

−→ Λk+1(Ω)

Formally: Given f ∈ Λk , find u ∈ Λk such that

(dk−1δk−1 + δkdk)u = f .

Here δk is a formal adjoint of dk .

The following mixed formulation is always well-posed: Givenf ∈ L2Λk(Ω), find σ ∈ HΛk−1, u ∈ HΛk and p ∈ Hk such that

〈σ, τ〉 − 〈dτ, u〉 = 0 ∀τ ∈ HΛk−1

〈dσ, v〉+ 〈du, dv〉+ 〈p, v〉 = 〈f , v〉 ∀v ∈ HΛk

〈u, q〉 = 0 ∀q ∈ Hk

Page 77: Introduction to finite element exterior calculus

Hodge LaplacianWell-posedness of the Hodge Laplace problem follows from theHodge decomposition and Poincare’s inequality:

‖ω‖L2 ≤ c ‖dω‖L2 , ω ∈ (Zk)⊥.

Special cases:Recall that if dim Hk = 0: Find σ ∈ HΛk−1, and u ∈ HΛk

〈σ, τ〉 − 〈dτ, u〉 = 0 ∀τ ∈ HΛk−1

〈dσ, v〉+ 〈du, dv〉 = 〈f , v〉 ∀v ∈ HΛk

I k = 0: ordinary LaplacianI k = n: mixed LaplacianI k = 1, n = 3: σ = − div u, gradσ + curl curl u = fI k = 2, n = 3: σ = curl u, curl σ − grad div u = f ,

Page 78: Introduction to finite element exterior calculus

Hodge LaplacianWell-posedness of the Hodge Laplace problem follows from theHodge decomposition and Poincare’s inequality:

‖ω‖L2 ≤ c ‖dω‖L2 , ω ∈ (Zk)⊥.

Special cases:Recall that if dim Hk = 0: Find σ ∈ HΛk−1, and u ∈ HΛk

〈σ, τ〉 − 〈dτ, u〉 = 0 ∀τ ∈ HΛk−1

〈dσ, v〉+ 〈du, dv〉 = 〈f , v〉 ∀v ∈ HΛk

I k = 0: ordinary LaplacianI k = n: mixed LaplacianI k = 1, n = 3: σ = − div u, gradσ + curl curl u = fI k = 2, n = 3: σ = curl u, curl σ − grad div u = f ,

Page 79: Introduction to finite element exterior calculus

Discretizations of the de Rham complex in threedimensions

R →H1(Ω)grad−−→ H(curl,Ω)

curl−−→ H(div,Ω)div−−→ L2(Ω) −→ 0yI1

h

yIch

yIdh

yI0h

R → H1h

grad−−→ Hh(curl)curl−−→ Hh(div)

div−−→ L2h −→ 0.

The operators Ih are the canonical interpolation operators definedfrom the degrees of freedom. They commute with grad curl anddiv, but the are not bounded on the spaces H1, H(curl) or H(div).

Page 80: Introduction to finite element exterior calculus

Discretizations of the de Rham complex in threedimensions

R →H1(Ω)grad−−→ H(curl,Ω)

curl−−→ H(div,Ω)div−−→ L2(Ω) −→ 0yI1

h

yIch

yIdh

yI0h

R → H1h

grad−−→ Hh(curl)curl−−→ Hh(div)

div−−→ L2h −→ 0.

The operators Ih are the canonical interpolation operators definedfrom the degrees of freedom. They commute with grad curl anddiv, but the are not bounded on the spaces H1, H(curl) or H(div).

Page 81: Introduction to finite element exterior calculus

Discretizations of the de Rham complex in n dimensions

R →HΛ0(Ω)d−→ HΛ1(Ω)

d−→ · · · d−→ HΛn(Ω) −→ 0.yIh yIh yIhR → Λ0

hd−→ Λ1

hd−→ · · · d−→ Λn

h −→ 0.

The operators Ih are the canonical interpolation operators definedfrom the degrees of freedom. They commute with d , but the arenot bounded on the HΛ.

Page 82: Introduction to finite element exterior calculus

Discretizations of the de Rham complex in n dimensions

R →HΛ0(Ω)d−→ HΛ1(Ω)

d−→ · · · d−→ HΛn(Ω) −→ 0.yIh yIh yIhR → Λ0

hd−→ Λ1

hd−→ · · · d−→ Λn

h −→ 0.

The operators Ih are the canonical interpolation operators definedfrom the degrees of freedom. They commute with d , but the arenot bounded on the HΛ.

Page 83: Introduction to finite element exterior calculus

Discretization, Abstract setting

· · · −→ Λk−1 dk−1

−−→ Λk −→ · · ·

x∪ x∪· · · −→ Λk−1

hdk−1

−−→ Λkh −→ · · ·

Complex of Hilbert spaces with dk bounded and closed range.

Hodge decomposition and Poincare inequality follow.

For discretization, construct a finite dimensional subcomplex.

Define Hkh = (Bk

h)⊥ ∩ Zkh .

Discrete Hodge decomposition follows: Λkh = Bk

h ⊕ Hkh ⊕ (Zk

h)⊥

Galerkin’s method: Λk−1, Λk , Hk −→ Λk−1h , Λk

h , Hkh

When is it stable?

Page 84: Introduction to finite element exterior calculus

Discretization, Abstract setting

· · · −→ Λk−1 dk−1

−−→ Λk −→ · · ·

x∪ x∪· · · −→ Λk−1

hdk−1

−−→ Λkh −→ · · ·

Complex of Hilbert spaces with dk bounded and closed range.

Hodge decomposition and Poincare inequality follow.

For discretization, construct a finite dimensional subcomplex.

Define Hkh = (Bk

h)⊥ ∩ Zkh .

Discrete Hodge decomposition follows: Λkh = Bk

h ⊕ Hkh ⊕ (Zk

h)⊥

Galerkin’s method: Λk−1, Λk , Hk −→ Λk−1h , Λk

h , Hkh

When is it stable?

Page 85: Introduction to finite element exterior calculus

Discretization, Abstract setting

· · · −→ Λk−1 dk−1

−−→ Λk −→ · · ·x∪ x∪· · · −→ Λk−1

hdk−1

−−→ Λkh −→ · · ·

Complex of Hilbert spaces with dk bounded and closed range.

Hodge decomposition and Poincare inequality follow.

For discretization, construct a finite dimensional subcomplex.

Define Hkh = (Bk

h)⊥ ∩ Zkh .

Discrete Hodge decomposition follows: Λkh = Bk

h ⊕ Hkh ⊕ (Zk

h)⊥

Galerkin’s method: Λk−1, Λk , Hk −→ Λk−1h , Λk

h , Hkh

When is it stable?

Page 86: Introduction to finite element exterior calculus

Discretization, Abstract setting

· · · −→ Λk−1 dk−1

−−→ Λk −→ · · ·x∪ x∪· · · −→ Λk−1

hdk−1

−−→ Λkh −→ · · ·

Complex of Hilbert spaces with dk bounded and closed range.

Hodge decomposition and Poincare inequality follow.

For discretization, construct a finite dimensional subcomplex.

Define Hkh = (Bk

h)⊥ ∩ Zkh .

Discrete Hodge decomposition follows: Λkh = Bk

h ⊕ Hkh ⊕ (Zk

h)⊥

Galerkin’s method: Λk−1, Λk , Hk −→ Λk−1h , Λk

h , Hkh

When is it stable?

Page 87: Introduction to finite element exterior calculus

Discretization, Abstract setting

· · · −→ Λk−1 dk−1

−−→ Λk −→ · · ·x∪ x∪· · · −→ Λk−1

hdk−1

−−→ Λkh −→ · · ·

Complex of Hilbert spaces with dk bounded and closed range.

Hodge decomposition and Poincare inequality follow.

For discretization, construct a finite dimensional subcomplex.

Define Hkh = (Bk

h)⊥ ∩ Zkh .

Discrete Hodge decomposition follows: Λkh = Bk

h ⊕ Hkh ⊕ (Zk

h)⊥

Galerkin’s method: Λk−1, Λk , Hk −→ Λk−1h , Λk

h , Hkh

When is it stable?

Page 88: Introduction to finite element exterior calculus

Discretization, Abstract setting

· · · −→ Λk−1 dk−1

−−→ Λk −→ · · ·x∪ x∪· · · −→ Λk−1

hdk−1

−−→ Λkh −→ · · ·

Complex of Hilbert spaces with dk bounded and closed range.

Hodge decomposition and Poincare inequality follow.

For discretization, construct a finite dimensional subcomplex.

Define Hkh = (Bk

h)⊥ ∩ Zkh .

Discrete Hodge decomposition follows: Λkh = Bk

h ⊕ Hkh ⊕ (Zk

h)⊥

Galerkin’s method: Λk−1, Λk , Hk −→ Λk−1h , Λk

h , Hkh

When is it stable?

Page 89: Introduction to finite element exterior calculus

Bounded cochain projections

Key property: Suppose that there exists a bounded cochainprojection.

· · · −→ Λk−1 dk−1

−−−→ Λk −→ · · ·yπk−1h

yπkh

· · · −→ Λk−1h

dk−1

−−−→ Λkh −→ · · ·

I πkh uniformly bounded

I ‖πkhω − ω‖ → 0.

I πkh a projection

I πkh dk−1 = dk−1πk−1

h

Theorem

I If ‖v − πkhv‖ < ‖v‖ ∀v ∈ Hk , then the induced map on

cohomology is an isomorphism.

I gap(Hk ,Hk

h

)≤ sup

v∈Hk ,‖v‖=1

‖v − πkhv‖

I The discrete Poincare inequality holds uniformly in h.

I Galerkin’s method is stable and convergent.

Page 90: Introduction to finite element exterior calculus

Bounded cochain projections

Key property: Suppose that there exists a bounded cochainprojection.

· · · −→ Λk−1 dk−1

−−−→ Λk −→ · · ·yπk−1h

yπkh

· · · −→ Λk−1h

dk−1

−−−→ Λkh −→ · · ·

I πkh uniformly bounded

I ‖πkhω − ω‖ → 0.

I πkh a projection

I πkh dk−1 = dk−1πk−1

h

Theorem

I If ‖v − πkhv‖ < ‖v‖ ∀v ∈ Hk , then the induced map on

cohomology is an isomorphism.

I gap(Hk ,Hk

h

)≤ sup

v∈Hk ,‖v‖=1

‖v − πkhv‖

I The discrete Poincare inequality holds uniformly in h.

I Galerkin’s method is stable and convergent.

Page 91: Introduction to finite element exterior calculus

Proof of discrete Poincare inequality

Theorem: There is a positive constant c , independent of h, suchthat

‖ω‖ ≤ c‖dω‖, ω ∈ Zk⊥h .

In fact, c ≤ cPcπ, where cP is continuous Poincare constant, andcπ bounds ‖πh‖.

Proof: Given ω ∈ Zk⊥h , define η ∈ Zk⊥ ⊂ HΛk(Ω) by dη = dω. By

the Poincare inequality, ‖η‖ ≤ c‖dω‖, so it is enough to show that‖ω‖ ≤ c‖η‖. Now, ω − πhη ∈ Λk

h and d(ω − πhη) = 0, soω − πhη ∈ Zk

h . Therefore

‖ω‖2 = 〈ω, πhη〉+ 〈ω, ω − πhη〉 = 〈ω, πhη〉 ≤ ‖ω‖‖πhη‖,

whence ‖ω‖ ≤ ‖πhη‖. The result follows from the uniformboundedness of πh.

Page 92: Introduction to finite element exterior calculus

Proof of discrete Poincare inequality

Theorem: There is a positive constant c , independent of h, suchthat

‖ω‖ ≤ c‖dω‖, ω ∈ Zk⊥h .

In fact, c ≤ cPcπ, where cP is continuous Poincare constant, andcπ bounds ‖πh‖.Proof: Given ω ∈ Zk⊥

h , define η ∈ Zk⊥ ⊂ HΛk(Ω) by dη = dω. Bythe Poincare inequality, ‖η‖ ≤ c‖dω‖, so it is enough to show that‖ω‖ ≤ c‖η‖. Now, ω − πhη ∈ Λk

h and d(ω − πhη) = 0, soω − πhη ∈ Zk

h . Therefore

‖ω‖2 = 〈ω, πhη〉+ 〈ω, ω − πhη〉 = 〈ω, πhη〉 ≤ ‖ω‖‖πhη‖,

whence ‖ω‖ ≤ ‖πhη‖. The result follows from the uniformboundedness of πh.

Page 93: Introduction to finite element exterior calculus

Construction of bounded cochain projections

The canonical projections, Ih, determined by the degrees offreedom, commute with d . But they are not bounded on HΛk .

If we apply the three operations:

I extend (E )

I regularize (R)

I canonical projection (Ih)

we get a map Qkh : HΛk(Ω)→ Λk

h which is bounded and commuteswith d . But it is not a projection.

However the composition

πkh = (Qk

h |Λkh)−1 Qk

h

can be shown to be a bounded cochain projection.

Page 94: Introduction to finite element exterior calculus

Construction of bounded cochain projections

The canonical projections, Ih, determined by the degrees offreedom, commute with d . But they are not bounded on HΛk .

If we apply the three operations:

I extend (E )

I regularize (R)

I canonical projection (Ih)

we get a map Qkh : HΛk(Ω)→ Λk

h which is bounded and commuteswith d . But it is not a projection.

However the composition

πkh = (Qk

h |Λkh)−1 Qk

h

can be shown to be a bounded cochain projection.

Page 95: Introduction to finite element exterior calculus

Construction of bounded cochain projections

The canonical projections, Ih, determined by the degrees offreedom, commute with d . But they are not bounded on HΛk .

If we apply the three operations:

I extend (E )

I regularize (R)

I canonical projection (Ih)

we get a map Qkh : HΛk(Ω)→ Λk

h which is bounded and commuteswith d . But it is not a projection.

However the composition

πkh = (Qk

h |Λkh)−1 Qk

h

can be shown to be a bounded cochain projection.

Page 96: Introduction to finite element exterior calculus

Bounded projections, definitionsWe combine extension and smoothing into an operatorRε

h : L2Λk(Ω)→ C Λk(Ω) given by

(Rεhω)x =

∫B1

ρ(y)(Eω)x+εhy dy .

Assume that the mesh is quasi–uniform:

Lemma

I For each ε > 0 sufficiently small there is a constant c(ε) suchthat for all h

‖IhRεh‖L(L2Λk (Ω),L2Λk (Ω)) ≤ c(ε).

I There is constant c, independent of ε and h such that

‖I − IhRεh‖L(L2Λk

h ,L2Λk

h) ≤ cε.

Page 97: Introduction to finite element exterior calculus

Bounded projections, definitionsWe combine extension and smoothing into an operatorRε

h : L2Λk(Ω)→ C Λk(Ω) given by

(Rεhω)x =

∫B1

ρ(y)(Eω)x+εhy dy .

Assume that the mesh is quasi–uniform:

Lemma

I For each ε > 0 sufficiently small there is a constant c(ε) suchthat for all h

‖IhRεh‖L(L2Λk (Ω),L2Λk (Ω)) ≤ c(ε).

I There is constant c, independent of ε and h such that

‖I − IhRεh‖L(L2Λk

h ,L2Λk

h) ≤ cε.

Page 98: Introduction to finite element exterior calculus

Scaling and compactness

Page 99: Introduction to finite element exterior calculus

Bounded cochain projections

The operator πh = πkh defined by

πh = (IhRεh|Λk

h)−1 IhRε

h,

where ε is taken small, but not too small, has all the desiredproperties, i.e.

I πkh is bounded in L(L2Λk(Ω), L2Λk(Ω)) andL(HΛk(Ω),HΛk(Ω))

I πkh commutes with the exterior derivative

I πkh is a projection

Page 100: Introduction to finite element exterior calculus

Removing the assumption of a quasi–uniform mesh

If the family of meshes is shape regular, but not necessarilyquasi–uniform, we introduce a piecewise linear and uniformlyLipschitz continuous function gh(x) to represent the local meshsize.

For each vertex x we let gh(x) be the average of the values ofdiamT for the simplices meeting x , and we defineΦεy

h (x) = x + εgh(x)y .

Then we replace the operator

(Rεhω)x =

∫B1

ρ(y)(Eω)x+εhy dy

by

(Rεhω)x =

∫B1

ρ(y)((Φεyh )∗Eω)x dy .

Page 101: Introduction to finite element exterior calculus

Removing the assumption of a quasi–uniform mesh

If the family of meshes is shape regular, but not necessarilyquasi–uniform, we introduce a piecewise linear and uniformlyLipschitz continuous function gh(x) to represent the local meshsize.

For each vertex x we let gh(x) be the average of the values ofdiamT for the simplices meeting x , and we defineΦεy

h (x) = x + εgh(x)y .

Then we replace the operator

(Rεhω)x =

∫B1

ρ(y)(Eω)x+εhy dy

by

(Rεhω)x =

∫B1

ρ(y)((Φεyh )∗Eω)x dy .

Page 102: Introduction to finite element exterior calculus

Removing the assumption of a quasi–uniform mesh

If the family of meshes is shape regular, but not necessarilyquasi–uniform, we introduce a piecewise linear and uniformlyLipschitz continuous function gh(x) to represent the local meshsize.

For each vertex x we let gh(x) be the average of the values ofdiamT for the simplices meeting x , and we defineΦεy

h (x) = x + εgh(x)y .

Then we replace the operator

(Rεhω)x =

∫B1

ρ(y)(Eω)x+εhy dy

by

(Rεhω)x =

∫B1

ρ(y)((Φεyh )∗Eω)x dy .

Page 103: Introduction to finite element exterior calculus

References

The development of FEEC leans heavily on earlier results takenfrom

I Whitney, Bossavit, Raviart and Thomas, Nedelec, Hiptmair,...

as well as on the theory of finite elements in general.

The presentation here is mostly based on

I D.N. Arnold, R.S. Falk, R. Winther, Finite element exteriorcalculus, homological techniques, and applications, ActaNumerica 2006.

I D.N. Arnold, R.S. Falk and R. Winther Finite element exteriorcalculus: From Hodge theory to numerical stability Bulletin ofthe Amer. Math. Soc. 47 (2010), 281-354.

Page 104: Introduction to finite element exterior calculus

References

The development of FEEC leans heavily on earlier results takenfrom

I Whitney, Bossavit, Raviart and Thomas, Nedelec, Hiptmair,...

as well as on the theory of finite elements in general.The presentation here is mostly based on

I D.N. Arnold, R.S. Falk, R. Winther, Finite element exteriorcalculus, homological techniques, and applications, ActaNumerica 2006.

I D.N. Arnold, R.S. Falk and R. Winther Finite element exteriorcalculus: From Hodge theory to numerical stability Bulletin ofthe Amer. Math. Soc. 47 (2010), 281-354.