finite element exterior calculus and its applications · finite element exterior calculus and its...
TRANSCRIPT
![Page 1: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/1.jpg)
Finite Element Exterior Calculus and ItsApplications
Douglas N. Arnold
Institute for Mathematics and its ApplicationsUniversity of Minnesota
July 31, 2006
![Page 2: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/2.jpg)
Outline
1 Introduction and motivating examples
2 Roots and ingredients of FEEC
3 Applications related to the Hodge Laplacian
4 Application to elasticity via BGG
![Page 3: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/3.jpg)
Main reference
DNA, R. Falk, R. Winther: Finite element exterior calculus,homological techniques, and applications, Acta Numerica 15(2006), pp. 1–155.
![Page 4: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/4.jpg)
Introduction
A great strength of finite element methods is that they oftenadmit a mathematical convergence theory, allowing validationand comparison of methods.
Approximability, consistency, and stability =⇒ convergence
Stability, like its continuous analogue, well-posedness, can beextremely subtle
Well-posedness + approximability + consistency 6=⇒stability
Exterior calculus, Hodge theory, de Rham cohomology, . . . ,were developed to get at well-posedness.FEEC adapts these tools to the discrete level to get at stability.
![Page 5: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/5.jpg)
Introduction
A great strength of finite element methods is that they oftenadmit a mathematical convergence theory, allowing validationand comparison of methods.
Approximability, consistency, and stability =⇒ convergence
Stability, like its continuous analogue, well-posedness, can beextremely subtle
Well-posedness + approximability + consistency 6=⇒stability
Exterior calculus, Hodge theory, de Rham cohomology, . . . ,were developed to get at well-posedness.FEEC adapts these tools to the discrete level to get at stability.
![Page 6: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/6.jpg)
Introduction
A great strength of finite element methods is that they oftenadmit a mathematical convergence theory, allowing validationand comparison of methods.
Approximability, consistency, and stability =⇒ convergence
Stability, like its continuous analogue, well-posedness, can beextremely subtle
Well-posedness + approximability + consistency 6=⇒stability
Exterior calculus, Hodge theory, de Rham cohomology, . . . ,were developed to get at well-posedness.FEEC adapts these tools to the discrete level to get at stability.
![Page 7: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/7.jpg)
Introduction
A great strength of finite element methods is that they oftenadmit a mathematical convergence theory, allowing validationand comparison of methods.
Approximability, consistency, and stability =⇒ convergence
Stability, like its continuous analogue, well-posedness, can beextremely subtle
Well-posedness + approximability + consistency 6=⇒stability
Exterior calculus, Hodge theory, de Rham cohomology, . . . ,were developed to get at well-posedness.FEEC adapts these tools to the discrete level to get at stability.
![Page 8: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/8.jpg)
Introduction
A great strength of finite element methods is that they oftenadmit a mathematical convergence theory, allowing validationand comparison of methods.
Approximability, consistency, and stability =⇒ convergence
Stability, like its continuous analogue, well-posedness, can beextremely subtle
Well-posedness + approximability + consistency 6=⇒stability
Exterior calculus, Hodge theory, de Rham cohomology, . . . ,were developed to get at well-posedness.FEEC adapts these tools to the discrete level to get at stability.
![Page 9: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/9.jpg)
Example 1: Laplacian
σ + u′ = 0, σ′ = f on (−1, 1) , u(±1) = 0
σ ∈ H1, u ∈ L2 :
∫στ =
∫uτ ′ ∀τ ∈ H1,
∫σ′v =
∫fv ∀v ∈ L2.
In higher dimensions, the solution is not obvious!2D: Raviart–Thomas ’76; Brezzi–Douglas–Marini ’85; 3D: Nedelec ’86
![Page 10: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/10.jpg)
Example 1: Laplacian
σ + u′ = 0, σ′ = f on (−1, 1) , u(±1) = 0
σ ∈ H1, u ∈ L2 :
∫στ =
∫uτ ′ ∀τ ∈ H1,
∫σ′v =
∫fv ∀v ∈ L2.
In higher dimensions, the solution is not obvious!2D: Raviart–Thomas ’76; Brezzi–Douglas–Marini ’85; 3D: Nedelec ’86
![Page 11: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/11.jpg)
Example 1: Laplacian
σ + u′ = 0, σ′ = f on (−1, 1) , u(±1) = 0
σ ∈ H1, u ∈ L2 :
∫στ =
∫uτ ′ ∀τ ∈ H1,
∫σ′v =
∫fv ∀v ∈ L2.
In higher dimensions, the solution is not obvious!2D: Raviart–Thomas ’76; Brezzi–Douglas–Marini ’85; 3D: Nedelec ’86
![Page 12: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/12.jpg)
Example 1: Laplacian
σ + u′ = 0, σ′ = f on (−1, 1) , u(±1) = 0
σ ∈ H1, u ∈ L2 :
∫στ =
∫uτ ′ ∀τ ∈ H1,
∫σ′v =
∫fv ∀v ∈ L2.
In higher dimensions, the solution is not obvious!2D: Raviart–Thomas ’76; Brezzi–Douglas–Marini ’85; 3D: Nedelec ’86
![Page 13: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/13.jpg)
Example 1: Laplacian
σ + u′ = 0, σ′ = f on (−1, 1) , u(±1) = 0
σ ∈ H1, u ∈ L2 :
∫στ =
∫uτ ′ ∀τ ∈ H1,
∫σ′v =
∫fv ∀v ∈ L2.
In higher dimensions, the solution is not obvious!2D: Raviart–Thomas ’76; Brezzi–Douglas–Marini ’85; 3D: Nedelec ’86
![Page 14: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/14.jpg)
Ex. 2a: Maxwell eigenvalue problem, unstructured mesh
∫Ω
curl u · curl v = λ
∫Ω
u · v ∀v
λ = m2 + n2 = 0, 1, 1, 2, 4, 4, 5, 5, 8, . . .
Boffi
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Ex. 2a: Maxwell eigenvalue problem, unstructured mesh
∫Ω
curl u · curl v = λ
∫Ω
u · v ∀v
λ = m2 + n2 = 0, 1, 1, 2, 4, 4, 5, 5, 8, . . .
Boffi
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Ex. 2a: Maxwell eigenvalue problem, unstructured mesh
∫Ω
curl u · curl v = λ
∫Ω
u · v ∀v
λ = m2 + n2 = 0, 1, 1, 2, 4, 4, 5, 5, 8, . . .
Boffi
![Page 17: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/17.jpg)
Ex. 2b: Maxwell eigenvalue problem, regular mesh
λ = m2 + n2 = 1, 1, 2, 4, 4, 5, 5, 8, . . .
254 574 1022 1598
1.0043 1.0019 1.0011 1.00071.0043 1.0019 1.0011 1.00072.0171 2.0076 2.0043 2.00274.0680 4.0304 4.0171 4.01104.0680 4.0304 4.0171 4.01105.1063 5.0475 5.0267 5.01715.1063 5.0475 5.0267 5.01715.9229 5.9658 5.9807 5.98778.2713 8.1215 8.0685 8.0438
Boffi-Brezzi-Gastaldi
![Page 18: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/18.jpg)
Ex. 2b: Maxwell eigenvalue problem, regular mesh
λ = m2 + n2 = 1, 1, 2, 4, 4, 5, 5, 8, . . .
254 574 1022 1598
1.0043 1.0019 1.0011 1.00071.0043 1.0019 1.0011 1.00072.0171 2.0076 2.0043 2.00274.0680 4.0304 4.0171 4.01104.0680 4.0304 4.0171 4.01105.1063 5.0475 5.0267 5.01715.1063 5.0475 5.0267 5.01715.9229 5.9658 5.9807 5.98778.2713 8.1215 8.0685 8.0438
Boffi-Brezzi-Gastaldi
![Page 19: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/19.jpg)
Outline
1 Introduction and motivating examples
2 Roots and ingredients of FEEC
3 Applications related to the Hodge Laplacian
4 Application to elasticity via BGG
![Page 20: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/20.jpg)
De Rham’s Theorem and . . .
Ω ⊂ Rn, HΛk(Ω) = ω ∈ L2Λk(Ω) | dω ∈ L2Λk+1(Ω)
0→ HΛ0(Ω)d−−→ HΛ1(Ω)
d−−→ · · · d−−→ HΛn(Ω)→ 0
If Ω is furnished with a simplicial decomposition T , a
many-to-one correspondence Λk(Ω)→ C ∗k (T ) (space of
k-cochains) is given by
ω 7→ (c 7→∫c ω) (?)
By Stokes theoremit’s a cochain map, soinduces a map fromde Rham to simplicialcohomology.
Λk(Ω)d−−→ Λk+1(Ω)y y
C∗k (T )∂∗−−→ C∗k+1(T )
De Rham’s thm: induced map is an isomorphism on cohomology.
![Page 21: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/21.jpg)
De Rham’s Theorem and . . .
Ω ⊂ Rn, HΛk(Ω) = ω ∈ L2Λk(Ω) | dω ∈ L2Λk+1(Ω)
0→ HΛ0(Ω)d−−→ HΛ1(Ω)
d−−→ · · · d−−→ HΛn(Ω)→ 0
If Ω is furnished with a simplicial decomposition T , a
many-to-one correspondence Λk(Ω)→ C ∗k (T ) (space of
k-cochains) is given by
ω 7→ (c 7→∫c ω) (?)
By Stokes theoremit’s a cochain map, soinduces a map fromde Rham to simplicialcohomology.
Λk(Ω)d−−→ Λk+1(Ω)y y
C∗k (T )∂∗−−→ C∗k+1(T )
De Rham’s thm: induced map is an isomorphism on cohomology.
![Page 22: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/22.jpg)
De Rham’s Theorem and . . .
Ω ⊂ Rn, HΛk(Ω) = ω ∈ L2Λk(Ω) | dω ∈ L2Λk+1(Ω)
0→ HΛ0(Ω)d−−→ HΛ1(Ω)
d−−→ · · · d−−→ HΛn(Ω)→ 0
If Ω is furnished with a simplicial decomposition T , a
many-to-one correspondence Λk(Ω)→ C ∗k (T ) (space of
k-cochains) is given by
ω 7→ (c 7→∫c ω) (?)
By Stokes theoremit’s a cochain map, soinduces a map fromde Rham to simplicialcohomology.
Λk(Ω)d−−→ Λk+1(Ω)y y
C∗k (T )∂∗−−→ C∗k+1(T )
De Rham’s thm: induced map is an isomorphism on cohomology.
![Page 23: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/23.jpg)
De Rham’s Theorem and . . .
Ω ⊂ Rn, HΛk(Ω) = ω ∈ L2Λk(Ω) | dω ∈ L2Λk+1(Ω)
0→ HΛ0(Ω)d−−→ HΛ1(Ω)
d−−→ · · · d−−→ HΛn(Ω)→ 0
If Ω is furnished with a simplicial decomposition T , a
many-to-one correspondence Λk(Ω)→ C ∗k (T ) (space of
k-cochains) is given by
ω 7→ (c 7→∫c ω) (?)
By Stokes theoremit’s a cochain map, soinduces a map fromde Rham to simplicialcohomology.
Λk(Ω)d−−→ Λk+1(Ω)y y
C∗k (T )∂∗−−→ C∗k+1(T )
De Rham’s thm: induced map is an isomorphism on cohomology.
![Page 24: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/24.jpg)
De Rham’s Theorem and The Roots of FEEC
Whitney ’57 constructed a cochain map C ∗k (T )→ HΛk(Ω)
which is a one-sided inverse to (?). Its range consists ofcertain piecewise linear k-forms (P−1 Λk(T ) in my notation). Inthis way the simplicial cochain complex is identified with asubcomplex of the de Rham complex:
0→ P−1 Λ0(T )d−−→ P−1 Λ1(T )
d−−→ · · · d−−→ P−1 Λn(T )→ 0
P−1 (T ) = P1(T ), all continuous piecewise linear functionsP−n (T ) = P0(T ), all piecewise constants.
Bossavit ’88 observed that these spaces of Whitney formscoincided with the lowest order cases of mixed finite elementsdeveloped by Raviart–Thomas ’76 and Nedelec ’80 for 1-formsand 2-forms in 2D and 3D.
![Page 25: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/25.jpg)
De Rham’s Theorem and The Roots of FEEC
Whitney ’57 constructed a cochain map C ∗k (T )→ HΛk(Ω)
which is a one-sided inverse to (?). Its range consists ofcertain piecewise linear k-forms (P−1 Λk(T ) in my notation). Inthis way the simplicial cochain complex is identified with asubcomplex of the de Rham complex:
0→ P−1 Λ0(T )d−−→ P−1 Λ1(T )
d−−→ · · · d−−→ P−1 Λn(T )→ 0
P−1 (T ) = P1(T ), all continuous piecewise linear functionsP−n (T ) = P0(T ), all piecewise constants.
Bossavit ’88 observed that these spaces of Whitney formscoincided with the lowest order cases of mixed finite elementsdeveloped by Raviart–Thomas ’76 and Nedelec ’80 for 1-formsand 2-forms in 2D and 3D.
![Page 26: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/26.jpg)
De Rham’s Theorem and The Roots of FEEC
Whitney ’57 constructed a cochain map C ∗k (T )→ HΛk(Ω)
which is a one-sided inverse to (?). Its range consists ofcertain piecewise linear k-forms (P−1 Λk(T ) in my notation). Inthis way the simplicial cochain complex is identified with asubcomplex of the de Rham complex:
0→ P−1 Λ0(T )d−−→ P−1 Λ1(T )
d−−→ · · · d−−→ P−1 Λn(T )→ 0
P−1 (T ) = P1(T ), all continuous piecewise linear functionsP−n (T ) = P0(T ), all piecewise constants.
Bossavit ’88 observed that these spaces of Whitney formscoincided with the lowest order cases of mixed finite elementsdeveloped by Raviart–Thomas ’76 and Nedelec ’80 for 1-formsand 2-forms in 2D and 3D.
![Page 27: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/27.jpg)
Finite Element de Rham subcomplexes
This is the fundamental structure of FEEC.
A finite element subcomplex of the de Rham complex
together with a bounded cochain projection
0 −−→ HΛ0(Ω)d−−→ HΛ1(Ω)
d−−→ · · · d−−→ HΛn(Ω) −−→ 0
π0
y π1
y πn
y
0 −−→ Λ0(T )d−−→ Λ1(T )
d−−→ · · · d−−→ Λn(T ) −−→ 0
The Λk(T ) are finite element spaces in the sense that they can beassembed from the following data on each simplex:
finite dimensional space of polynomials forms on the simplex, and
a decomposition of its dual space into subspaces associated to thesubsimplices (degrees of freedom)
![Page 28: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/28.jpg)
Finite Element de Rham subcomplexes
This is the fundamental structure of FEEC.
A finite element subcomplex of the de Rham complex
together with a bounded cochain projection
0 −−→ HΛ0(Ω)d−−→ HΛ1(Ω)
d−−→ · · · d−−→ HΛn(Ω) −−→ 0
π0
y π1
y πn
y0 −−→ Λ0(T )
d−−→ Λ1(T )d−−→ · · · d−−→ Λn(T ) −−→ 0
The Λk(T ) are finite element spaces in the sense that they can beassembed from the following data on each simplex:
finite dimensional space of polynomials forms on the simplex, and
a decomposition of its dual space into subspaces associated to thesubsimplices (degrees of freedom)
![Page 29: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/29.jpg)
Finite Element de Rham subcomplexes
This is the fundamental structure of FEEC.
A finite element subcomplex of the de Rham complex
together with a bounded cochain projection
0 −−→ HΛ0(Ω)d−−→ HΛ1(Ω)
d−−→ · · · d−−→ HΛn(Ω) −−→ 0
π0
y π1
y πn
y0 −−→ Λ0(T )
d−−→ Λ1(T )d−−→ · · · d−−→ Λn(T ) −−→ 0
The Λk(T ) are finite element spaces in the sense that they can beassembed from the following data on each simplex:
finite dimensional space of polynomials forms on the simplex, and
a decomposition of its dual space into subspaces associated to thesubsimplices (degrees of freedom)
![Page 30: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/30.jpg)
Construction of FE differential forms
The key to the construction is the Koszul differentialκ : Λk → Λk−1:
(κω)x(v1, . . . , vk−1) = ωx(x , v1, . . . , vk−1)
κ : PrΛk → Pr+1Λ
k−1 (c.f. d : Pr+1Λk−1 → PrΛ
k)
0 ←−− PrΛ0 κ←−− Pr−1Λ
1 κ←−− · · · κ←−− Pr−nΛn ← 0
Koszul complex
(dκ + κd)ω = (r + k)ω ∀ω ∈ HrΛk (homogeneous
polynomials)
κ is a contracting chain homotopy
HrΛk = dHr+1Λ
k−1 ⊕ κHr−1Λk+1
![Page 31: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/31.jpg)
Construction of FE differential forms
The key to the construction is the Koszul differentialκ : Λk → Λk−1:
(κω)x(v1, . . . , vk−1) = ωx(x , v1, . . . , vk−1)
κ : PrΛk → Pr+1Λ
k−1 (c.f. d : Pr+1Λk−1 → PrΛ
k)
0 ←−− PrΛ0 κ←−− Pr−1Λ
1 κ←−− · · · κ←−− Pr−nΛn ← 0
Koszul complex
(dκ + κd)ω = (r + k)ω ∀ω ∈ HrΛk (homogeneous
polynomials)
κ is a contracting chain homotopy
HrΛk = dHr+1Λ
k−1 ⊕ κHr−1Λk+1
![Page 32: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/32.jpg)
Construction of FE differential forms
The key to the construction is the Koszul differentialκ : Λk → Λk−1:
(κω)x(v1, . . . , vk−1) = ωx(x , v1, . . . , vk−1)
κ : PrΛk → Pr+1Λ
k−1 (c.f. d : Pr+1Λk−1 → PrΛ
k)
0 ←−− PrΛ0 κ←−− Pr−1Λ
1 κ←−− · · · κ←−− Pr−nΛn ← 0
Koszul complex
(dκ + κd)ω = (r + k)ω ∀ω ∈ HrΛk (homogeneous
polynomials)
κ is a contracting chain homotopy
HrΛk = dHr+1Λ
k−1 ⊕ κHr−1Λk+1
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Construction of FE differential forms
The key to the construction is the Koszul differentialκ : Λk → Λk−1:
(κω)x(v1, . . . , vk−1) = ωx(x , v1, . . . , vk−1)
κ : PrΛk → Pr+1Λ
k−1 (c.f. d : Pr+1Λk−1 → PrΛ
k)
0 ←−− PrΛ0 κ←−− Pr−1Λ
1 κ←−− · · · κ←−− Pr−nΛn ← 0
Koszul complex
(dκ + κd)ω = (r + k)ω ∀ω ∈ HrΛk (homogeneous
polynomials)
κ is a contracting chain homotopy
HrΛk = dHr+1Λ
k−1 ⊕ κHr−1Λk+1
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Construction of FE differential forms
The key to the construction is the Koszul differentialκ : Λk → Λk−1:
(κω)x(v1, . . . , vk−1) = ωx(x , v1, . . . , vk−1)
κ : PrΛk → Pr+1Λ
k−1 (c.f. d : Pr+1Λk−1 → PrΛ
k)
0 ←−− PrΛ0 κ←−− Pr−1Λ
1 κ←−− · · · κ←−− Pr−nΛn ← 0
Koszul complex
(dκ + κd)ω = (r + k)ω ∀ω ∈ HrΛk (homogeneous
polynomials)
κ is a contracting chain homotopy
HrΛk = dHr+1Λ
k−1 ⊕ κHr−1Λk+1
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PrΛk and P−r Λk
Using the Koszul differential, we define a special space ofpolynomial differential k-forms between PrΛ
k and Pr−1Λk :
Pr
−
Λk := Pr−1Λk + κHr−1Λ
k+1 + dHr+1Λk−1
X
Note that P−r Λ0 = PrΛ0 and P−r Λn = Pr−1Λ
n
God made PrΛk and P−r Λk ,
all the rest is the work of man.
Proven with representation theory. . .
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PrΛk and P−r Λk
Using the Koszul differential, we define a special space ofpolynomial differential k-forms between PrΛ
k and Pr−1Λk :
Pr−Λk := Pr−1Λ
k + κHr−1Λk+1 + dHr+1Λ
k−1X
Note that P−r Λ0 = PrΛ0 and P−r Λn = Pr−1Λ
n
God made PrΛk and P−r Λk ,
all the rest is the work of man.
Proven with representation theory. . .
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PrΛk and P−r Λk
Using the Koszul differential, we define a special space ofpolynomial differential k-forms between PrΛ
k and Pr−1Λk :
Pr−Λk := Pr−1Λ
k + κHr−1Λk+1 + dHr+1Λ
k−1X
Note that P−r Λ0 = PrΛ0 and P−r Λn = Pr−1Λ
n
God made PrΛk and P−r Λk ,
all the rest is the work of man.
Proven with representation theory. . .
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PrΛk and P−r Λk
Using the Koszul differential, we define a special space ofpolynomial differential k-forms between PrΛ
k and Pr−1Λk :
Pr−Λk := Pr−1Λ
k + κHr−1Λk+1 + dHr+1Λ
k−1X
Note that P−r Λ0 = PrΛ0 and P−r Λn = Pr−1Λ
n
God made PrΛk and P−r Λk ,
all the rest is the work of man.
Proven with representation theory. . .
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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
(PrΛk(T ))∗ and (P−r Λk(T ))∗ (T a simplex), into subspaces associated
to subsimplices f of T .
DOF for PrΛk(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−1Λ
d−k(f ) Hiptmair
Given a triangulation T , we can then define PrΛk(T ), P−r Λk(T ).
They are subspaces of HΛk(Ω).
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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
(PrΛk(T ))∗ and (P−r Λk(T ))∗ (T a simplex), into subspaces associated
to subsimplices f of T .
DOF for PrΛk(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−1Λ
d−k(f ) Hiptmair
Given a triangulation T , we can then define PrΛk(T ), P−r Λk(T ).
They are subspaces of HΛk(Ω).
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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
(PrΛk(T ))∗ and (P−r Λk(T ))∗ (T a simplex), into subspaces associated
to subsimplices f of T .
DOF for PrΛk(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−1Λ
d−k(f ) Hiptmair
Given a triangulation T , we can then define PrΛk(T ), P−r Λk(T ).
They are subspaces of HΛk(Ω).
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Finite element differential forms/Mixed FEM
P−r Λ0(T ) = PrΛ0(T ) ⊂ H1 Lagrange elts
P−r Λn(T ) = Pr−1Λn(T ) ⊂ L2 discontinuous elts
n = 2: P−r Λ1(T ) ⊂ H(curl) Raviart–Thomas elts
n = 2:
Pr Λ1(T ) ⊂ H(curl) Brezzi–Douglas–Marini elts
n = 3: P−r Λ1(T ) ⊂ H(curl) Nedelec 1st kind edge elts
n = 3:
Pr Λ1(T ) ⊂ H(curl) Nedelec 2nd kind edge elts
n = 3:
P−r Λ2(T ) ⊂ H(div) Nedelec 1st kind face elts
n = 3:
Pr Λ2(T ) ⊂ H(div) Nedelec 2nd kind face elts
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Finite element de Rham subcomplexes
For every r ≥ 1, the P−r Λk spaces give a FE de Rham subcomplex:
0→ P−r Λ0(T )d−−→ P−r Λ1(T )
d−−→ · · · d−−→ P−r Λn(T )→ 0
For r = 1 this is Whitney’s complex.
The projections Πk : Λk(Ω)→ P−r Λk(T ) defined through theDOF form a cochain projection. (They are not defined on allof HΛk(Ω) but modified cochain projections can be definedwhich are bounded on HΛk .)
There are many ways to form the spaces PrΛk(T ) and
P−r Λk(T ) into a discrete de Rham subcomplex with a cochainprojection:there are 2n−1 such sequences for each value of r .
In every case the cochain projection induces an isomorphismon cohomology.
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Finite element de Rham subcomplexes
For every r ≥ 1, the P−r Λk spaces give a FE de Rham subcomplex:
0→ P−r Λ0(T )d−−→ P−r Λ1(T )
d−−→ · · · d−−→ P−r Λn(T )→ 0
For r = 1 this is Whitney’s complex.
The projections Πk : Λk(Ω)→ P−r Λk(T ) defined through theDOF form a cochain projection. (They are not defined on allof HΛk(Ω) but modified cochain projections can be definedwhich are bounded on HΛk .)
There are many ways to form the spaces PrΛk(T ) and
P−r Λk(T ) into a discrete de Rham subcomplex with a cochainprojection:there are 2n−1 such sequences for each value of r .
In every case the cochain projection induces an isomorphismon cohomology.
![Page 45: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/45.jpg)
Finite element de Rham subcomplexes
For every r ≥ 1, the P−r Λk spaces give a FE de Rham subcomplex:
0→ P−r Λ0(T )d−−→ P−r Λ1(T )
d−−→ · · · d−−→ P−r Λn(T )→ 0
For r = 1 this is Whitney’s complex.
The projections Πk : Λk(Ω)→ P−r Λk(T ) defined through theDOF form a cochain projection. (They are not defined on allof HΛk(Ω) but modified cochain projections can be definedwhich are bounded on HΛk .)
There are many ways to form the spaces PrΛk(T ) and
P−r Λk(T ) into a discrete de Rham subcomplex with a cochainprojection:there are 2n−1 such sequences for each value of r .
In every case the cochain projection induces an isomorphismon cohomology.
![Page 46: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/46.jpg)
Finite element de Rham subcomplexes
For every r ≥ 1, the P−r Λk spaces give a FE de Rham subcomplex:
0→ P−r Λ0(T )d−−→ P−r Λ1(T )
d−−→ · · · d−−→ P−r Λn(T )→ 0
For r = 1 this is Whitney’s complex.
The projections Πk : Λk(Ω)→ P−r Λk(T ) defined through theDOF form a cochain projection. (They are not defined on allof HΛk(Ω) but modified cochain projections can be definedwhich are bounded on HΛk .)
There are many ways to form the spaces PrΛk(T ) and
P−r Λk(T ) into a discrete de Rham subcomplex with a cochainprojection:there are 2n−1 such sequences for each value of r .
In every case the cochain projection induces an isomorphismon cohomology.
![Page 47: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/47.jpg)
Outline
1 Introduction and motivating examples
2 Roots and ingredients of FEEC
3 Applications related to the Hodge Laplacian
4 Application to elasticity via BGG
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Mixed Hodge Laplacian
Ω ⊂ Rn, 0 ≤ k ≤ n, f ∈ L2Λk(Ω)
σ ∈ HΛk−1(Ω), u ∈ HΛk(Ω) :
〈σ, τ〉 − 〈dτ, u〉 = 0 ∀τ ∈ HΛk−1(Ω)
〈dσ, v〉+ 〈du, dv〉 = 〈f , v〉 ∀v ∈ HΛk(Ω)
k = 0: ordinary Laplacian
k = n: mixed Laplacian
k = 1, n = 3: σ = − div u, grad σ + curl curl u = f
k = 2, n = 3: σ = curl u, curlσ − grad div u = f
For special f these reduce to
div u = f , curl u = 0
curl curl u = f , div u = 0
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Mixed Hodge Laplacian
Ω ⊂ Rn, 0 ≤ k ≤ n, f ∈ L2Λk(Ω)
σ ∈ HΛk−1(Ω), u ∈ HΛk(Ω) :
〈σ, τ〉 − 〈dτ, u〉 = 0 ∀τ ∈ HΛk−1(Ω)
〈dσ, v〉+ 〈du, dv〉 = 〈f , v〉 ∀v ∈ HΛk(Ω)
k = 0: ordinary Laplacian
k = n: mixed Laplacian
k = 1, n = 3: σ = − div u, grad σ + curl curl u = f
k = 2, n = 3: σ = curl u, curlσ − grad div u = f
For special f these reduce to
div u = f , curl u = 0
curl curl u = f , div u = 0
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Mixed Hodge Laplacian
Ω ⊂ Rn, 0 ≤ k ≤ n, f ∈ L2Λk(Ω)
σ ∈ HΛk−1(Ω), u ∈ HΛk(Ω) :
〈σ, τ〉 − 〈dτ, u〉 = 0 ∀τ ∈ HΛk−1(Ω)
〈dσ, v〉+ 〈du, dv〉 = 〈f , v〉 ∀v ∈ HΛk(Ω)
k = 0: ordinary Laplacian
k = n: mixed Laplacian
k = 1, n = 3: σ = − div u, grad σ + curl curl u = f
k = 2, n = 3: σ = curl u, curlσ − grad div u = f
For special f these reduce to
div u = f , curl u = 0
curl curl u = f , div u = 0
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Well-posedness of the Hodge Laplacian
To obtain well-posedness we must handle the harmonic forms
hk := u ∈ HΛk | du = 0, 〈dτ, u〉 = 0 ∀τ ∈ HΛk−1
σ ∈ HΛk−1, u ∈ HΛk , p ∈ hk :
〈σ, τ〉 − 〈dτ, u〉 = 0 ∀τ ∈ HΛk−1
〈dσ, v〉+ 〈du, dv〉+ 〈p, v〉 = 〈f , v〉 ∀v ∈ HΛk
〈u, q〉 = 0 ∀q ∈ hk
Need to control ‖σ‖HΛ + ‖u‖HΛ + ‖p‖ by a bounded choice of τ , v , and
q.
τ = σ controls ‖σ‖, v = p controls ‖p‖, v = dσ controls‖dσ‖v = u controls ‖du‖, How to control ‖u‖??
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Well-posedness of the Hodge Laplacian
To obtain well-posedness we must handle the harmonic forms
hk := u ∈ HΛk | du = 0, 〈dτ, u〉 = 0 ∀τ ∈ HΛk−1
σ ∈ HΛk−1, u ∈ HΛk , p ∈ hk :
〈σ, τ〉 − 〈dτ, u〉 = 0 ∀τ ∈ HΛk−1
〈dσ, v〉+ 〈du, dv〉+ 〈p, v〉 = 〈f , v〉 ∀v ∈ HΛk
〈u, q〉 = 0 ∀q ∈ hk
Need to control ‖σ‖HΛ + ‖u‖HΛ + ‖p‖ by a bounded choice of τ , v , and
q.
τ = σ controls ‖σ‖, v = p controls ‖p‖, v = dσ controls‖dσ‖v = u controls ‖du‖, How to control ‖u‖??
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Well-posedness of the Hodge Laplacian
To obtain well-posedness we must handle the harmonic forms
hk := u ∈ HΛk | du = 0, 〈dτ, u〉 = 0 ∀τ ∈ HΛk−1
σ ∈ HΛk−1, u ∈ HΛk , p ∈ hk :
〈σ, τ〉 − 〈dτ, u〉 = 0 ∀τ ∈ HΛk−1
〈dσ, v〉+ 〈du, dv〉+ 〈p, v〉 = 〈f , v〉 ∀v ∈ HΛk
〈u, q〉 = 0 ∀q ∈ hk
Need to control ‖σ‖HΛ + ‖u‖HΛ + ‖p‖ by a bounded choice of τ , v , and
q.
τ = σ controls ‖σ‖, v = p controls ‖p‖, v = dσ controls‖dσ‖v = u controls ‖du‖
, How to control ‖u‖??
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Well-posedness of the Hodge Laplacian
To obtain well-posedness we must handle the harmonic forms
hk := u ∈ HΛk | du = 0, 〈dτ, u〉 = 0 ∀τ ∈ HΛk−1
σ ∈ HΛk−1, u ∈ HΛk , p ∈ hk :
〈σ, τ〉 − 〈dτ, u〉 = 0 ∀τ ∈ HΛk−1
〈dσ, v〉+ 〈du, dv〉+ 〈p, v〉 = 〈f , v〉 ∀v ∈ HΛk
〈u, q〉 = 0 ∀q ∈ hk
Need to control ‖σ‖HΛ + ‖u‖HΛ + ‖p‖ by a bounded choice of τ , v , and
q.
τ = σ controls ‖σ‖, v = p controls ‖p‖, v = dσ controls‖dσ‖v = u controls ‖du‖, How to control ‖u‖??
![Page 55: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/55.jpg)
Well-posedness from the Hodge decomposition
HΛk−1 d−→ HΛk d−→ HΛk+1
u ∈ HΛk = dHΛk−1 ⊕ (dHΛk−1)⊥
Since dHΛk−1 ⊕ hk = N (d), (dHΛk−1)⊥ = hk ⊕N (d)⊥. Thus
HΛk = dHΛk−1 ⊕ hk ⊕N (d)⊥ Hodge decomposition
u = dτ + q + z , τ ∈ HΛk−1, q ∈ hk , z ∈ N (d)⊥
Easy to bound ‖dτ‖ and ‖q‖. To bound ‖z‖ we use Poincare’sinequality ‖z‖ ≤ c‖dz‖ for z ∈ N (d)⊥, and the fact that dz = du,which is already under control.
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Well-posedness from the Hodge decomposition
HΛk−1 d−→ HΛk d−→ HΛk+1
u ∈ HΛk = dHΛk−1 ⊕ (dHΛk−1)⊥
Since dHΛk−1 ⊕ hk = N (d), (dHΛk−1)⊥ = hk ⊕N (d)⊥. Thus
HΛk = dHΛk−1 ⊕ hk ⊕N (d)⊥ Hodge decomposition
u = dτ + q + z , τ ∈ HΛk−1, q ∈ hk , z ∈ N (d)⊥
Easy to bound ‖dτ‖ and ‖q‖. To bound ‖z‖ we use Poincare’sinequality ‖z‖ ≤ c‖dz‖ for z ∈ N (d)⊥, and the fact that dz = du,which is already under control.
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Stability of the FE for the mixed Hodge Laplacian
Analogous reasoning using the finite element de Rham complex,establishes stability of the finite element. In place of the Poincareinequality we use the Poincare inequality on the continuous leveland the boundedness of the cochain projections. A full convergencetheory follows for four different families of mixed finite elements!
P−r Λk−1(T )× P−r Λk(T )
PrΛk−1(T )× P−r Λk(T )
P−r+1Λk−1(T )× PrΛ
k(T )
Pr+1Λk−1(T )× PrΛ
k(T )
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There are lots of other applications of FEEC
Maxwell’s equations and related EM problems
Mixed eigenvalue problems
Preconditioning and multigrid
Stable mixed FEM for elasticity
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Outline
1 Introduction and motivating examples
2 Roots and ingredients of FEEC
3 Applications related to the Hodge Laplacian
4 Application to elasticity via BGG
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The equations of elasticity
Ω ⊂ R3, f : Ω→ R3 imposed load.Find stress σ : Ω→ R3×3
sym , displacement u : Ω→ R3 such that
Aσ = ε(u), div σ = f
Finding stable finite elements for this first order system is a longopen, very challenging, and very important problem.
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The equations of elasticity
Ω ⊂ R3, f : Ω→ R3 imposed load.Find stress σ : Ω→ R3×3
sym , displacement u : Ω→ R3 such that
Aσ = ε(u), div σ = f
Finding stable finite elements for this first order system is a longopen, very challenging, and very important problem.
![Page 62: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/62.jpg)
The elasticity complex
For the equations of elasticity, the relevant elliptic complex is
0→ C∞(Ω, R3)ε−−→ C∞(Ω, R3×3
sym )J−−→ C∞(Ω, R3×3
sym )div−−→ C∞(Ω, R3)→ 0
↑displacement
↑strain
↑stress
↑load
Jτ = curl(curl τ)T , second order
With weakly imposed symmetry the relevant sequence is
0→C∞(R3×R3×3skw )
(grad,−I )−−−−−→C∞(R3×3)J−→C∞(R3×3)
0@ div
skw
1A−−−−→C∞(R3 × R3×3
skw )→0
where J is extended by zero to skew matrices.
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The elasticity complex
For the equations of elasticity, the relevant elliptic complex is
0→ C∞(Ω, R3)ε−−→ C∞(Ω, R3×3
sym )J−−→ C∞(Ω, R3×3
sym )div−−→ C∞(Ω, R3)→ 0
↑displacement
↑strain
↑stress
↑load
Jτ = curl(curl τ)T , second order
With weakly imposed symmetry the relevant sequence is
0→C∞(R3×R3×3skw )
(grad,−I )−−−−−→C∞(R3×3)J−→C∞(R3×3)
0@ div
skw
1A−−−−→C∞(R3 × R3×3
skw )→0
where J is extended by zero to skew matrices.
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The elasticity complex
For the equations of elasticity, the relevant elliptic complex is
0→ C∞(Ω, R3)ε−−→ C∞(Ω, R3×3
sym )J−−→ C∞(Ω, R3×3
sym )div−−→ C∞(Ω, R3)→ 0
↑displacement
↑strain
↑stress
↑load
Jτ = curl(curl τ)T , second order
With weakly imposed symmetry the relevant sequence is
0→C∞(R3×R3×3skw )
(grad,−I )−−−−−→C∞(R3×3)J−→C∞(R3×3)
0@ div
skw
1A−−−−→C∞(R3 × R3×3
skw )→0
where J is extended by zero to skew matrices.
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Bernstein–Gelfand–Gelfand construction, I
V = Rn, K = V ∧ V, W = K× V.1. Start with the de Rham sequence with values in W:
0→Λ0(Ω; W)
(d 00 d
)−−−→ Λ1(Ω; W)
(d 00 d
)−−−→ · · ·
(d 00 d
)−−−→ Λn(Ω; W)→ 0
2. Define K : Λk(V)→ Λk(K) by
(Kω)x(v1, . . . , vk) = x ∧ ωx(v1, . . . , vk)
3. Define automorphisms Φ : Λk(W)→ Λk(W) by
Φ =
(I K0 I
), Φ−1 =
(I −K0 I
)
4. Define A = Φ (
d 00 d
)Φ−1 to get a modified de Rham sequence:
0→Λ0(W)A−−→ Λ1(W)
A−−→ · · · A−−→ Λn(W)→ 0
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Bernstein–Gelfand–Gelfand construction, I
V = Rn, K = V ∧ V, W = K× V.1. Start with the de Rham sequence with values in W:
0→Λ0(Ω; W)
(d 00 d
)−−−→ Λ1(Ω; W)
(d 00 d
)−−−→ · · ·
(d 00 d
)−−−→ Λn(Ω; W)→ 0
2. Define K : Λk(V)→ Λk(K) by
(Kω)x(v1, . . . , vk) = x ∧ ωx(v1, . . . , vk)
3. Define automorphisms Φ : Λk(W)→ Λk(W) by
Φ =
(I K0 I
), Φ−1 =
(I −K0 I
)
4. Define A = Φ (
d 00 d
)Φ−1 to get a modified de Rham sequence:
0→Λ0(W)A−−→ Λ1(W)
A−−→ · · · A−−→ Λn(W)→ 0
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Bernstein–Gelfand–Gelfand construction, I
V = Rn, K = V ∧ V, W = K× V.1. Start with the de Rham sequence with values in W:
0→Λ0(Ω; W)
(d 00 d
)−−−→ Λ1(Ω; W)
(d 00 d
)−−−→ · · ·
(d 00 d
)−−−→ Λn(Ω; W)→ 0
2. Define K : Λk(V)→ Λk(K) by
(Kω)x(v1, . . . , vk) = x ∧ ωx(v1, . . . , vk)
3. Define automorphisms Φ : Λk(W)→ Λk(W) by
Φ =
(I K0 I
), Φ−1 =
(I −K0 I
)
4. Define A = Φ (
d 00 d
)Φ−1 to get a modified de Rham sequence:
0→Λ0(W)A−−→ Λ1(W)
A−−→ · · · A−−→ Λn(W)→ 0
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Bernstein–Gelfand–Gelfand construction, I
V = Rn, K = V ∧ V, W = K× V.1. Start with the de Rham sequence with values in W:
0→Λ0(Ω; W)
(d 00 d
)−−−→ Λ1(Ω; W)
(d 00 d
)−−−→ · · ·
(d 00 d
)−−−→ Λn(Ω; W)→ 0
2. Define K : Λk(V)→ Λk(K) by
(Kω)x(v1, . . . , vk) = x ∧ ωx(v1, . . . , vk)
3. Define automorphisms Φ : Λk(W)→ Λk(W) by
Φ =
(I K0 I
), Φ−1 =
(I −K0 I
)
4. Define A = Φ (
d 00 d
)Φ−1 to get a modified de Rham sequence:
0→Λ0(W)A−−→ Λ1(W)
A−−→ · · · A−−→ Λn(W)→ 0
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Bernstein–Gelfand–Gelfand construction, I
V = Rn, K = V ∧ V, W = K× V.1. Start with the de Rham sequence with values in W:
0→Λ0(Ω; W)
(d 00 d
)−−−→ Λ1(Ω; W)
(d 00 d
)−−−→ · · ·
(d 00 d
)−−−→ Λn(Ω; W)→ 0
2. Define K : Λk(V)→ Λk(K) by
(Kω)x(v1, . . . , vk) = x ∧ ωx(v1, . . . , vk)
3. Define automorphisms Φ : Λk(W)→ Λk(W) by
Φ =
(I K0 I
), Φ−1 =
(I −K0 I
)
4. Define A = Φ (
d 00 d
)Φ−1 to get a modified de Rham sequence:
0→Λ0(W)A−−→ Λ1(W)
A−−→ · · · A−−→ Λn(W)→ 0
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BGG II
5. Note that A =
(d −S0 d
), where S = dK − Kd : Λk(V)→ Λk+1(K)
is given by
(Sω)x(v1, . . . , vk+1) =k+1∑j=1
(−1)j+1vj ∧ ωx(v1, . . . , vj , . . . , vk+1).
Properties: S is algebraic. For k = n − 2, S is an isomorphism. dS = −Sd .
6. Define subspaces Γk ⊂ Λk(W) satisfying A(Γk) ⊂ Γk+1 and projections
πk : Λk(W)→ Γk satisfying πk+1A = Aπk :
Γn−2 = (ω, µ) ∈ Λn−2(W) : dω = Sµ , Γn−1 = (ω, µ) ∈ Λn−1(W) : ω = 0
πn−2 =
„I 0
S−1d 0
«: Λn−2(W) → Γn−2, πn−1 =
„0 0
dS−1 I
«: Λn−1(W) → Γn−1.
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BGG II
5. Note that A =
(d −S0 d
), where S = dK − Kd : Λk(V)→ Λk+1(K)
is given by
(Sω)x(v1, . . . , vk+1) =k+1∑j=1
(−1)j+1vj ∧ ωx(v1, . . . , vj , . . . , vk+1).
Properties: S is algebraic. For k = n − 2, S is an isomorphism. dS = −Sd .
6. Define subspaces Γk ⊂ Λk(W) satisfying A(Γk) ⊂ Γk+1 and projections
πk : Λk(W)→ Γk satisfying πk+1A = Aπk :
Γn−2 = (ω, µ) ∈ Λn−2(W) : dω = Sµ , Γn−1 = (ω, µ) ∈ Λn−1(W) : ω = 0
πn−2 =
„I 0
S−1d 0
«: Λn−2(W) → Γn−2, πn−1 =
„0 0
dS−1 I
«: Λn−1(W) → Γn−1.
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BGG II
5. Note that A =
(d −S0 d
), where S = dK − Kd : Λk(V)→ Λk+1(K)
is given by
(Sω)x(v1, . . . , vk+1) =k+1∑j=1
(−1)j+1vj ∧ ωx(v1, . . . , vj , . . . , vk+1).
Properties: S is algebraic. For k = n − 2, S is an isomorphism. dS = −Sd .
6. Define subspaces Γk ⊂ Λk(W) satisfying A(Γk) ⊂ Γk+1 and projections
πk : Λk(W)→ Γk satisfying πk+1A = Aπk :
Γn−2 = (ω, µ) ∈ Λn−2(W) : dω = Sµ , Γn−1 = (ω, µ) ∈ Λn−1(W) : ω = 0
πn−2 =
„I 0
S−1d 0
«: Λn−2(W) → Γn−2, πn−1 =
„0 0
dS−1 I
«: Λn−1(W) → Γn−1.
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BGG III
7. The following diagram with vertical projections commutes (dS = −Sd):
· · · →Λn−3(W)A−−→ Λn−2(W)
A−−→ Λn−1(W)A−−→ Λn(W)→ 0yid
yπn−2
yπn−1
yid
· · · →Λn−3(W)A−−→ Γn−2 A−−→ Γn−1 A−−→ Λn(W)→ 0
Therefore, the subcomplex on the bottom row is exact if the top is.
8. This subcomplex may be identified with the elasticity complex.
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BGG III
7. The following diagram with vertical projections commutes (dS = −Sd):
· · · →Λn−3(W)A−−→ Λn−2(W)
A−−→ Λn−1(W)A−−→ Λn(W)→ 0yid
yπn−2
yπn−1
yid
· · · →Λn−3(W)A−−→ Γn−2 A−−→ Γn−1 A−−→ Λn(W)→ 0
Therefore, the subcomplex on the bottom row is exact if the top is.
8. This subcomplex may be identified with the elasticity complex.
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BGG III
7. The following diagram with vertical projections commutes (dS = −Sd):
· · · →Λn−3(W)A−−→ Λn−2(W)
A−−→ Λn−1(W)A−−→ Λn(W)→ 0yid
yπn−2
yπn−1
yid
· · · →Λn−3(W)A−−→ Γn−2 A−−→ Γn−1 A−−→ Λn(W)→ 0
Therefore, the subcomplex on the bottom row is exact if the top is.
8. This subcomplex may be identified with the elasticity complex.
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Mixed finite elements for elasticity
We can mimic the BGG construction on the discrete level.
We begin by picking two different finite element de Rhamsequences
· · · →Λn−3(T ) −−→ Λn−2(T ) −−→ Λn−1(T ) −−→ Λn(T ) −−→ 0
· · · →Λn−3(T ) −−→ Λn−2(T ) −−→ Λn−1(T ) −−→ Λn(T ) −−→ 0
Define KT = ΠT K : Λk(T ; V)→ Λk(T ; K),ST = dKT − KT d : Λk(T ; V)→ Λk+1(T ; K). For the discreteanalogue of the construction to go through, we make acompatibility requirement:for k = n − 2, ST is onto.If this holds, we finally conclude that the spaces Λn−1(T ; V) for σ,Λn(T ; V) for u, and Λn(T ; K) for p gives a stable discretization forelasticity.
![Page 77: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/77.jpg)
Mixed finite elements for elasticity
We can mimic the BGG construction on the discrete level.
We begin by picking two different finite element de Rhamsequences
· · · →Λn−3(T ) −−→ Λn−2(T ) −−→ Λn−1(T ) −−→ Λn(T ) −−→ 0
· · · →Λn−3(T ) −−→ Λn−2(T ) −−→ Λn−1(T ) −−→ Λn(T ) −−→ 0
Define KT = ΠT K : Λk(T ; V)→ Λk(T ; K),ST = dKT − KT d : Λk(T ; V)→ Λk+1(T ; K).
For the discreteanalogue of the construction to go through, we make acompatibility requirement:for k = n − 2, ST is onto.If this holds, we finally conclude that the spaces Λn−1(T ; V) for σ,Λn(T ; V) for u, and Λn(T ; K) for p gives a stable discretization forelasticity.
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Mixed finite elements for elasticity
We can mimic the BGG construction on the discrete level.
We begin by picking two different finite element de Rhamsequences
· · · →Λn−3(T ) −−→ Λn−2(T ) −−→ Λn−1(T ) −−→ Λn(T ) −−→ 0
· · · →Λn−3(T ) −−→ Λn−2(T ) −−→ Λn−1(T ) −−→ Λn(T ) −−→ 0
Define KT = ΠT K : Λk(T ; V)→ Λk(T ; K),ST = dKT − KT d : Λk(T ; V)→ Λk+1(T ; K). For the discreteanalogue of the construction to go through, we make acompatibility requirement:for k = n − 2, ST is onto.
If this holds, we finally conclude that the spaces Λn−1(T ; V) for σ,Λn(T ; V) for u, and Λn(T ; K) for p gives a stable discretization forelasticity.
![Page 79: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/79.jpg)
Mixed finite elements for elasticity
We can mimic the BGG construction on the discrete level.
We begin by picking two different finite element de Rhamsequences
· · · →Λn−3(T ) −−→ Λn−2(T ) −−→ Λn−1(T ) −−→ Λn(T ) −−→ 0
· · · →Λn−3(T ) −−→ Λn−2(T ) −−→ Λn−1(T ) −−→ Λn(T ) −−→ 0
Define KT = ΠT K : Λk(T ; V)→ Λk(T ; K),ST = dKT − KT d : Λk(T ; V)→ Λk+1(T ; K). For the discreteanalogue of the construction to go through, we make acompatibility requirement:for k = n − 2, ST is onto.If this holds, we finally conclude that the spaces Λn−1(T ; V) for σ,Λn(T ; V) for u, and Λn(T ; K) for p gives a stable discretization forelasticity.
![Page 80: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/80.jpg)
Stable elasticity elements
There are many pairs of finite element de Rham complexessatisfying the compability condition. The simplest is:
· · · →P−r+1Λn−3
(T ; K)
−→P−r+1Λn−2
(T ; K)
−→P−r+1Λn−1
(T ; K)
−→PrΛn
(T ; K)
→ 0
KT
x KT
x
ST
KT
x KT
x
· · · →Pr+2Λn−3
(T ; V)
−→P−r+2Λn−2
(T ; V)
−→Pr+1Λn−1
(T ; V)
−→PrΛn
(T ; V)
→ 0
They satisfy the compatibility condition because P−r+2Λn−2
includes face DOFs, needed for surjectivity of ST onto P−r+1Λn−1.
This choice leads to the following stable elements for elasticity:stress Pr+1Λ
n−1(T ; V)displacement PrΛ
n(T ; V)multiplier PrΛ
n(T ; K)
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Stable elasticity elements
There are many pairs of finite element de Rham complexessatisfying the compability condition. The simplest is:
· · · →P−r+1Λn−3(T ; K) −→P−r+1Λ
n−2(T ; K) −→P−r+1Λn−1(T ; K) −→PrΛ
n(T ; K)→ 0
KT
x KT
x
ST
KT
x KT
x
· · · →Pr+2Λn−3(T ; V) −→P−r+2Λ
n−2(T ; V) −→Pr+1Λn−1(T ; V) −→PrΛ
n(T ; V)→ 0
They satisfy the compatibility condition because P−r+2Λn−2
includes face DOFs, needed for surjectivity of ST onto P−r+1Λn−1.
This choice leads to the following stable elements for elasticity:stress Pr+1Λ
n−1(T ; V)displacement PrΛ
n(T ; V)multiplier PrΛ
n(T ; K)
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Stable elasticity elements
There are many pairs of finite element de Rham complexessatisfying the compability condition. The simplest is:
· · · →P−r+1Λn−3(T ; K) −→P−r+1Λ
n−2(T ; K) −→P−r+1Λn−1(T ; K) −→PrΛ
n(T ; K)→ 0
KT
x KT
x
ST
KT
x KT
x· · · →Pr+2Λ
n−3(T ; V) −→P−r+2Λn−2(T ; V) −→Pr+1Λ
n−1(T ; V) −→PrΛn(T ; V)→ 0
They satisfy the compatibility condition because P−r+2Λn−2
includes face DOFs, needed for surjectivity of ST onto P−r+1Λn−1.
This choice leads to the following stable elements for elasticity:stress Pr+1Λ
n−1(T ; V)displacement PrΛ
n(T ; V)multiplier PrΛ
n(T ; K)
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Stable elasticity elements
There are many pairs of finite element de Rham complexessatisfying the compability condition. The simplest is:
· · · →P−r+1Λn−3(T ; K) −→P−r+1Λ
n−2(T ; K) −→P−r+1Λn−1(T ; K) −→PrΛ
n(T ; K)→ 0
KT
x KT
x ST KT
x KT
x· · · →Pr+2Λ
n−3(T ; V) −→P−r+2Λn−2(T ; V) −→Pr+1Λ
n−1(T ; V) −→PrΛn(T ; V)→ 0
They satisfy the compatibility condition because P−r+2Λn−2
includes face DOFs, needed for surjectivity of ST onto P−r+1Λn−1.
This choice leads to the following stable elements for elasticity:stress Pr+1Λ
n−1(T ; V)displacement PrΛ
n(T ; V)multiplier PrΛ
n(T ; K)
![Page 84: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/84.jpg)
Stable elasticity elements
There are many pairs of finite element de Rham complexessatisfying the compability condition. The simplest is:
· · · →P−r+1Λn−3(T ; K) −→P−r+1Λ
n−2(T ; K) −→P−r+1Λn−1(T ; K) −→PrΛ
n(T ; K)→ 0
KT
x KT
x ST KT
x KT
x· · · →Pr+2Λ
n−3(T ; V) −→P−r+2Λn−2(T ; V) −→Pr+1Λ
n−1(T ; V) −→PrΛn(T ; V)→ 0
They satisfy the compatibility condition because P−r+2Λn−2
includes face DOFs, needed for surjectivity of ST onto P−r+1Λn−1.
This choice leads to the following stable elements for elasticity:stress Pr+1Λ
n−1(T ; V)displacement PrΛ
n(T ; V)multiplier PrΛ
n(T ; K)
![Page 85: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/85.jpg)
Stable elasticity elements
There are many pairs of finite element de Rham complexessatisfying the compability condition. The simplest is:
· · · →P−r+1Λn−3(T ; K) −→P−r+1Λ
n−2(T ; K) −→P−r+1Λn−1(T ; K) −→PrΛ
n(T ; K)→ 0
KT
x KT
x ST KT
x KT
x· · · →Pr+2Λ
n−3(T ; V) −→P−r+2Λn−2(T ; V) −→Pr+1Λ
n−1(T ; V) −→PrΛn(T ; V)→ 0
They satisfy the compatibility condition because P−r+2Λn−2
includes face DOFs, needed for surjectivity of ST onto P−r+1Λn−1.
This choice leads to the following stable elements for elasticity:stress Pr+1Λ
n−1(T ; V)displacement PrΛ
n(T ; V)multiplier PrΛ
n(T ; K)
![Page 86: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/86.jpg)
Simplest case
r = 0
Far simpler than the elements than any previously devised stablemixed elasticity elements.
![Page 87: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/87.jpg)
Conclusions
FEEC provides a very natural framework for the design andunderstanding of subtle stability issues that arise in thediscretization of a wide variety of PDE systems.
FEEC brings to bear tools from geometry, topology, andalgebra to develop discretizations which are compatible withthe geometric, topological, and algebraic structure of the PDEsystem, and so obtain stability.
FEEC has been used to unify, clarify, and refine many knownfinite element methods.
Via BGG FEEC has enabled major progress in thelong-standing problem of mixed discretizations of elasticity.
![Page 88: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/88.jpg)
Conclusions
FEEC provides a very natural framework for the design andunderstanding of subtle stability issues that arise in thediscretization of a wide variety of PDE systems.
FEEC brings to bear tools from geometry, topology, andalgebra to develop discretizations which are compatible withthe geometric, topological, and algebraic structure of the PDEsystem, and so obtain stability.
FEEC has been used to unify, clarify, and refine many knownfinite element methods.
Via BGG FEEC has enabled major progress in thelong-standing problem of mixed discretizations of elasticity.
![Page 89: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/89.jpg)
Conclusions
FEEC provides a very natural framework for the design andunderstanding of subtle stability issues that arise in thediscretization of a wide variety of PDE systems.
FEEC brings to bear tools from geometry, topology, andalgebra to develop discretizations which are compatible withthe geometric, topological, and algebraic structure of the PDEsystem, and so obtain stability.
FEEC has been used to unify, clarify, and refine many knownfinite element methods.
Via BGG FEEC has enabled major progress in thelong-standing problem of mixed discretizations of elasticity.
![Page 90: Finite Element Exterior Calculus and Its Applications · Finite Element Exterior Calculus and Its Applications Douglas N. Arnold Institute for Mathematics and its Applications University](https://reader030.vdocuments.us/reader030/viewer/2022021501/5ad2a51a7f8b9a482c8ca60e/html5/thumbnails/90.jpg)
Conclusions
FEEC provides a very natural framework for the design andunderstanding of subtle stability issues that arise in thediscretization of a wide variety of PDE systems.
FEEC brings to bear tools from geometry, topology, andalgebra to develop discretizations which are compatible withthe geometric, topological, and algebraic structure of the PDEsystem, and so obtain stability.
FEEC has been used to unify, clarify, and refine many knownfinite element methods.
Via BGG FEEC has enabled major progress in thelong-standing problem of mixed discretizations of elasticity.