Finite Element Exterior Calculus

Finite element exterior calculus

Douglas N. Arnold

School of Mathematics, University of Minnesota

NSF/CBMS Conference, ICERM, June 11–15, 2012

collaborators: R. Falk, R. Winther, G. Awanou, F. Bonizzoni, D. Bofﬁ, J. Gopalakrishnan, H. Chen Computational examples Standard P1 ﬁnite elements for 1D Laplacian

−u0 u

2 / 94 Mixed ﬁnite elements for Laplacian

σ u

0.25 0.0625

0.05 0.0312

u P1-P0 −0.15 u RT-P0 0 3 / 94 Vector Laplacian, L-shaped domain

curl curl u − grad div u = f in Ω u · n = 0, curl u × n = 0 on ∂Ω

Z Z (curl u · curl v + div u div v) = f · v ∀v Ω Ω

Lagrange ﬁnite elements converge nicely but not to the solution! (same problem with any conforming FE)

4 / 94 Vector Poisson equation

curl curl u − grad div u = f in Ω u · n = 0, curl u × n = 0 on ∂Ω

f ≡ 0 does not imply u ≡ 0:

dim H = b1 %- harmonic forms 1st Betti number (solutions for f = 0) (number of holes)

curl curl u − grad div u = f (mod H), u ⊥ H, b.c.

5 / 94 Maxwell eigenvalue problem

Find 0 6= u ∈ H(curl) s.t Z Z curl u · curl v = λ u · v ∀v ∈ H(curl) Ω Ω λ = m2 + n2 = 0, 1, 1, 2, 4, 4, 5, 5, 8,...

10 9 8 7 6 5 4 3 2 1 0

− 1 P1 P1 Λ

6 / 94 Maxwell eigenvalue problem, crisscross mesh

λ = m2 + n2 = 1, 1, 2, 4, 4, 5, 5, 8,...

254 574 1022 1598 10 9 1.0043 1.0019 1.0011 1.0007 8 1.0043 1.0019 1.0011 1.0007 7 2.0171 2.0076 2.0043 2.0027 6 5 4.0680 4.0304 4.0171 4.0110 4 4.0680 4.0304 4.0171 4.0110 3 5.1063 5.0475 5.0267 5.0171 2 5.1063 5.0475 5.0267 5.0171 1 5.9229 5.9658 5.9807 5.9877 0 8.2713 8.1215 8.0685 8.0438 Bofﬁ-Brezzi-Gastaldi ’99

7 / 94 EM calculations based on the generalized RT elements

Schoberl,¨ Zaglmayr 2006, NGSolve

− 1 2K tets, P6 Λ

8 / 94 Homology 101 Chain complexes

A chain complex (V , ∂) is a seq. of vector spaces and linear maps ∂k+1 ∂k k k+1 · · · → Vk+1 −−→ Vk −→ Vk−1 → · · · with ∂ ◦ ∂ = 0. typically, non-negative and ﬁnite: V k = 0 for k < 0 for k large L In other words, V = k Vk is a graded vector space and ∂ : V → V is a graded linear operator of degree −1 such that

∂ ◦ ∂ = 0( ∂k = ∂|Vk : Vk → Vk )

Vk : k-chains Zk = N (∂k ): k-cycles Bk = R(∂k+1): k-boundaries Hk = Zk /Bk : k-th homology space

Thus the elements of Hk are equivalence classes of k-cycles

9 / 94 Simplices and simplicial complexes

n A k-simplex in R is the convex hull f = [x0,..., xk ] of k + 1 vertices in general position.

A subset determines a face of f : [xi0 ,..., xid ].

Simplicial complex: A ﬁnite set S of simplices in Rn, such that 1. Faces of simplices in S are in S. 2. If f ∩ g 6= ∅ for f , g ∈ S, then it is a face of f and of g.

If we order all vertices of S, then an ordering of the vertices of the simplex determines an orientation.

1 2 3

0 +

2 0 0 1 0 1

10 / 94 The boundary operator on chains

∆k (S): the set of k-simplices in S X Ck (k-chains): formal linear combinations c = cf f

f ∈∆k (S) Pk i ∂k : ∆k → Ck−1: ∂[x0, x1,..., xk ] = i=0(−1) [..., xˆi ,...] P ∂k : Ck → Ck−1: ∂c = cf ∂f

11 / 94 The simplicial chain complex

∂ ∂ ∂ ∂ 0 → Cn −→ Cn−1 −→· · · −→ C0 −→ 0

βk := dim Hk (C) is the kth Betti number

1, 1, 0, 0 1, 1, 0, 0 1, 2, 1, 0 1, 2 , 0, 0 1, 0, 1, 0

12 / 94 Chain maps

∂k+1 ∂k · · · −→Vk+1 −−−→ Vk −→ Vk−1 −→· · ·    f   f  k+1y fk y k−1y ∂0 ∂0 0 k+1 0 k 0 · · · −→Vk+1 −−−→ Vk −→ Vk−1 −→· · ·

f (Z) ⊂ Z0, f (B) ⊂ B0, so f induces ¯f : H(V ) → H(V 0). 0 0 0 If V is a subcomplex (Vk ⊂ Vk and ∂ = ∂|V ), and fv = v for v ∈ V 0, we call f a chain projection.

Proposition A chain projection induces a surjection on homology.

13 / 94 Cochain complexes

A cochain complex is like a chain complex but with increasing indices.

dk−1 dk · · · → V k−1 −−−→ V k −→ V k+1 → · · ·

cocycles Zk , coboundaries Bk , cohomology Hk ,...

The dual of a chain complex is a cochain complex:

∗ ∗ ∗ ∂k+1 : Vk+1 → Vk =⇒ ∂k+1 : Vk → Vk+1

d k V k

14 / 94 The de Rham complex for a domain in Rn

d/dx 1-D: 0 → C∞(Ω) −−−→ C∞(Ω) → 0

∞ grad ∞ 2 rot ∞ 2-D: 0 → C (Ω) −−→ C (Ω, R ) −→ C (Ω) → 0

∞ grad ∞ 3 curl ∞ 3 div ∞ 3-D: 0 → C (Ω) −−→ C (Ω, R ) −−→ C (Ω, R ) −→ C (Ω) → 0

d d d d n-D: 0 → Λ0(Ω) −→ Λ1(Ω) −→ Λ2(Ω) −→· · · −→ Λn(Ω) → 0

k ∞ n×···×n The space Λ (Ω) = C (Ω, Rskw ), the space of smooth differential k-forms on Ω.

Exterior derivative:d k :Λk (Ω) → Λk+1(Ω) R Integral of a k-form over an oriented k-simplex: f v ∈ R R R k Stokes theorem: c du = ∂c u, u ∈ Λ , c ∈ Ck All this works on any smooth manifold

15 / 94 De Rham’s Theorem

k k ∗ De Rham map: Λ (Ω) −→ C (S) := Ck (S)

7−→ R u (c 7→ c u)

Stokes theorem d k d k+1 d says it’s a cochain ··· −→ Λ (Ω) −→ Λ (Ω) −→· · ·   map, so induces a   map from de Rham y y ∗ ∗ ∗ to simplicial cohomology. ∂ ∗ ∂ ∗ ∂ ··· −→ Ck −→ Ck+1 −→ · · ·

Theorem (De Rham’s theorem) The induced map is an isomorphism on cohomology.

16 / 94 Nonzero cohomology classes

1 1 u = grad θ, 0 6= u¯ ∈ H u = grad , 0 6= u¯ ∈ H2 r on cylindrical shell on spherical shell

17 / 94 Unbounded operators on Hilbert space Unbounded operators

X,Y H-spaces (extensions to Banach spaces, TVSs,. . . ) T : D(T ) → Y linear, D(T ) ⊆ X subspace (not necessarily closed), T not necessarily bounded Not-necessarily-everywhere-deﬁned-and-not-necessarily-bounded linear operators Densely deﬁned: D(T ) = X Ex: X = L2(Ω), Y = L2(Ω; Rn), D(T ) = H1(Ω), Tv = grad v (changing D(T ) to H˚1(Ω) gives a different example) S, T unbdd ops X → Y =⇒ D(S + T ) = D(S) ∩ D(T ) (may not be d.d.) S T X −→ Y , Y −→ Z unbdd ops =⇒ D(T ◦ S) = {v ∈ D(S) | Sv ∈ D(T )} 2 2 2 Graph norm (and inner product): kvkD(T ) := kvkX + kTvkY , v ∈ D(T ) Null space, range, graph: N (T ), R(T ), Γ(T )

18 / 94 Closed operators

T is closed if Γ(T ) is closed in X × Y . Equivalent deﬁnitions: X Y 1. If v1, v2,... ∈ D(T ) satisfy vn −→ x and Tvn −→ y for some x ∈ X and y ∈ Y , then x ∈ D(T ) and Tx = y. 2. D(T ) endowed with the graph norm is complete. If D(T ) = X, then T is closed ⇐⇒ T is bdd (Closed Graph Thm) Many properties of bounded operators extend to closed operators. E.g., Proposition Let T be a closed operator X to Y . 1. N (T ) is closed in X.

2. ∃γ > 0 s.t. kTxkY ≥ γkxkX ⇐⇒ N (T ) = 0, R(T ) closed 3. If dim Y / R(T ) < ∞, then R(T ) is closed

19 / 94 Adjoint of a d.d.unbdd operator

Let T be a d.d.unbdd operator X → Y . Deﬁne ∗ D(T ) = {w ∈ Y | the map v ∈ D(T ) 7→ hw, TviY ∈ R is bdd in X-norm } For w ∈ D(T ∗) ∃! T ∗w ∈ X s.t. ∗ ∗ hT w, viX = hw, TviY , v ∈ D(T ), w ∈ D(T ). T ∗ is a closed operator (even if T is not). Deﬁne the rotated graph Γ(˜ T ∗)= { (−T ∗w, w) | w ∈ D(T ∗) } ⊂ X × Y , Then Γ(T )⊥ = Γ(˜ T ∗), Γ(T ) = Γ(˜ T ∗)⊥. Proposition Let T be a closed d.d. operator X → Y . Then 1. T ∗ is closed d.d. 2. T ∗∗ = T. 3. R(T )⊥ = N (T ∗), N (T )⊥ = R(T ∗), R(T ∗)⊥ = N (T ), N (T ∗)⊥ = R(T ).

20 / 94 Closed Range Theorem

Theorem Let T be a closed d.d.operator X → Y . If R(T ) is closed in Y , then R(T ∗) is closed in X.

Proof. 1. Reduce to case T is surjective. 2. Restrict to orthog comp of N (T ) in D(T ) (w/ graph norm). Get bounded linear isomorphism. ∃ bounded inverse:

∀y ∈ Y ∃x ∈ X s.t. Tx = y, kxkX ≤ ckykY

∗ ∗ 3. This implies kykY ≤ ckT ykX , y ∈ D(T ).

21 / 94 Grad and div

Assume Ω ⊂ R3 with Lipschitz boundary (so trace theorem holds).

∞ 2 3 2 1. Start with − div : Cc ⊂ L (Ω; R ) → L (Ω)

2. Its adjoint is grad with domain H1 (this proves H1 is complete).

3. The adjoint of (grad, H1) is − div with domain

H˚(div) = {w ∈ H(div) | γnw := w · n|∂Ω = 0 }

4. div H˚(div) is ﬁnite-codimensional so closed. So grad H1 is closed.

22 / 94 Curl

∞ 2 3 2 3 1. Start with curl : Cc ⊂ L (Ω; R ) → L (Ω; R )

2. Its adjoint is curl with domain H(curl) (complete).

3. Adjoint of (curl, H(curl)) with domain

H˚(curl) = {w ∈ H(curl) | γt w := w × n|∂Ω = 0 }

4. We shall see that curl H(curl) is closed.

23 / 94 Hilbert complexes Hilbert complexes

Deﬁnition A Hilbert complex is a sequence of Hilbert spaces W k and a sequence of closed d.d.linear operators d k from W k to W k+1 such that R(d k ) ⊂ N (d k+1).

k 2 2 k 2 Vk = D(d ) H-space with graph norm: kvkV k = kvkW k + kd vkW k+1 The domain complex

d d d 0 → V 0 −→ V 1 −→· · · −→ V n → 0

is a bounded Hilbert complex (with less information). It is a cochain complex, so it has (co)cycles, boundaries, and homology. An H-complex is closed if Bk is closed in W k (or V k ). An H-complex is Fredholm if dim Hk < ∞.

Fredholm =⇒ closed

24 / 94 The dual complex

∗ ∗ k k−1 k−1 k k−1 k Deﬁne dk : Vk ⊂ W → W as the adjoint of d : V ⊂ W → W . It is closed d.d.and, since R(d k−1) ⊂ N (d k ),

∗ ∗ k ⊥ k−1 ⊥ k R(dk+1) ⊂ R(dk+1) = N (d ) ⊂ R(d ) = N (d∗ ),

so we get a Hilbert chain complex with domain complex

∗ d∗ ∗ n dn n−1 n−1 d1 ∗ 0 → V −→ V −−→· · · −→ V0 → 0.

If (W , d) is closed, then (W , d∗) is as well, by the Closed Range Theorem.

From now on we mainly deal with closed H-complexes. . .

25 / 94 Harmonic forms

The Hilbert structure of a closed H-complex allows us to identify the homology space Hk = Zk /Bk with a subspace Hk of W k :

k k k⊥ k ∗ k ∗ ∗ H := Z ∩ B = Z ∩ Zk = {u ∈ V ∩ Vk | du = 0, d u = 0}.

k ∗ An H-complex has the compactness property if V ∩ Vk is dense and compact in W k . This implies dim Hk < ∞.

compactness property =⇒ Fredholm =⇒ closed

26 / 94 Two key properties of closed H-complexes

Theorem (Hodge decomposition) For any closed Hilbert complex:

k k ¨ k ¨ ∗ W = B H Bk | {z } |{z} Zk Zk⊥ z }| { V k = Bk ¨ Hk ¨ Zk⊥V

Theorem (Poincare´ inequality) For any closed Hilbert complex, ∃ a constant cP s.t.

P k⊥V kzkV ≤ c kdzk, z ∈ Z .

27 / 94 L2 de Rham complex on Ω ⊂ R3

k k k ∗ ∗ k k W d V dk Vk dim H

2 1 2 0 L (Ω) grad H 0 L β0 2 3 ˚ 1 L (Ω; R ) curl H(curl) − div H(div) β1 2 3 ˚ 2 L (Ω; R ) div H(div) curl H(curl) β2 3 L2(Ω) 0 L2 − grad H˚1 0

grad curl div 0 → H1 −−→ H(curl) −−→ H(div) −→ L2 → 0

−div curl − grad 0 ← L2 ←−−− H˚(div) ←−− H˚(curl) ←−−−− H˚1 ← 0

28 / 94 The abstract Hodge Laplacian

d d k−1 k k+1 ∗ ∗ k L k W W W L := d d + dd W −→ W d∗ d∗ k k ∗ ∗ ∗ k−1 D(L ) = { u ∈ V ∩ Vk | du ∈ Vk+1, d u ∈ V } N (Lk ) = Hk , Hk ⊥ R(Lk ) k Strong formulation: Find u ∈ D(L ) s.t. Lu = f − PHf , u ⊥ H. k ∗ k⊥ Primal weak formulation: Find u ∈ V ∩ Vk ∩ H s.t.

∗ ∗ k ∗ k⊥ hdu, dvi + hd u, d vi = hf − PHf , vi, v ∈ V ∩ Vk ∩ H .

Mixed weak formulation. Find σ ∈ V k−1, u ∈ V k , and p ∈ Hk s.t.

hσ, τi − hu, dτi = 0, τ ∈ V k−1, hdσ, vi + hdu, dvi + hp, vi = hf , vi, v ∈ V k , hu, qi = 0, q ∈ Hk .

29 / 94 Equivalence and well-posedness

Theorem Let f ∈ W k . Then u ∈ W k solves the strong formulation ⇐⇒ it solves the primal weak formulation. Moreover, in this case, if we set ∗ σ = d u and p = PHu, then the triple (σ, u, p) solves the mixed weak formulation. Finally, if some (σ, u, p) solves the mixed weak ∗ formulation, then σ = d u, p = PHu, and u solves the strong and primal formulations of the problem.

Theorem For each f ∈ W k there exists a unique solution. Moreover

∗ ∗ ∗ kuk + kduk + kd uk + kdd uk + kd duk ≤ ckf − PHf k.

The constant depends only on the Poincare´ inequality constant cP .

30 / 94 Proof of well-posedness

We used the mixed formulation. Set

B(σ, u, p; τ, v, q) = hσ, τi−hu, dτi−hdσ, vi−hdu, dvi−hp, vi−hu, qi

We must prove the inf-sup condition: ∀ (σ, u, p) ∃ (τ, v, q) s.t.

B(σ, u, p; τ, v, q) ≥ γ(kσkV + kukV + kpk)(kτkV + kvkV + kqk),

with γ = γ(cP ) > 0. Via the Hodge decomposition,

u = uB + uH + uB∗ = dρ + uH + uB∗

with ρ ∈ Z⊥V . Then take

1 τ = σ − ρ, v = −u − dσ − p, q = p − uH. (cP )2

31 / 94 Hodge Laplacian and Hodge decomposition

∗ ∗ f = dd u + PHf + d du is the Hodge decomposition of f Deﬁne K : W k → D(Lk ) by Kf = u (bdd lin op). ∗ ∗ PB = dd K , PB∗ = d dK If f ∈ V , then Kdf = dKf . If f ∈ B, then dKf = 0. Since Kf ⊥ H, Kf ∈ B. B problem: If f ∈ B, then u = Kf solves

dd∗u = f , du = 0, u ⊥ H.

B∗ problem: If f ∈ B∗, then u = Kf solves

d∗du = f , d∗u = 0, u ⊥ H.

32 / 94 The Hodge Laplacian on a domain in 3D

grad curl div 0 → H1 −−→ H(curl) −−→ H(div) −→ L2 → 0

−div curl − grad 0 ← L2 ←−−− H˚(div) ←−− H˚(curl) ←−−−− H˚1 ← 0

k Lk = d∗d + dd∗ BCs imposed on. . . V k−1 × V k

0 −∆ ∂u/∂n H1

1 curl curl − grad div u · n curl u × n H1 × H(curl)

2 − grad div + curl curl u × n div u H(curl) × H(div)

3 −∆ u H(div) × L2

essential BC for primal form. natural BC for primal form.

33 / 94 Approximation of Hilbert complexes Why mixed methods?

Naively, we might try to discretize the primal formulation with ﬁnite elements. This works in some circumstances, but we have seen two ways in which it can fail. It is not easy to construct a dense family of k ∗ k subspaces of the primal energy space V ∩ Vk ∩ H .

We therefore consider ﬁnite element discretizations of the mixed formulation:

Given f ∈ W k , ﬁnd σ ∈ V k−1, u ∈ V k , and p ∈ Hk s.t.

hσ, τi − hu, dτi = 0,τ ∈ V k−1, hdσ, vi + hdu, dvi + hp, vi = hf , vi, v ∈ V k , hu, qi = 0, q ∈ Hk .

34 / 94 Galerkin method

j j Choose f.d. subspaces Vh ⊂ V j j j j j−1 j Zh = {v ∈ Vh|dv = 0} ⊂ Z Bh = {dv|v ∈ Vh } ⊂ B j j j Hh = {v ∈ Zh|v ⊥ Bh}

k k−1 k k Given f ∈ W , ﬁnd σh ∈ Vh , uh ∈ Vh , and ph ∈ Hh s.t.

k−1 hσh, τi − huh, dτi = 0,τ ∈ Vh , k hdσh, vi + hduh, dvi + hph, vi = hf , vi, v ∈ Vh , k huh, qi = 0, q ∈ Hh.

k k If Hh * H this is a nonconforming method.

j For any choice of the Vh there exists a unique solution. However, the consistency, stability, and accuracy of the discrete solution depends vitally on the choice of subspaces.

35 / 94 Key assumptions

j j We need the spaces Vh ⊂ V (at least for j = k − 1, k, k + 1) to satisfy three properties: j 1. Approximation property: Of course Vh must afford good approximation of elements of V j . This can be formalized with respect to a family of subspaces parametrized by h by requiring

j lim inf kw − vkV = 0, w ∈ V h→0 j v∈Vh

(or = O(hr ) for w in some dense subspace, or . . . ) k−1 k k k+1 2. Subcomplex property: dVh ⊂ Vh and dVh ⊂ Vh , so

k−1 d k d k+1 · · · → Vh −→ Vh −→ Vh → · · ·

is a subcomplex.

36 / 94 Bounded cochain projection

3. Bounded cochain projection: Most important, we assume that there exists a cochain map from the H-complex to the subcomplex which is a projection and is bounded.

d d V k−1 −→ V k −→ V k+1    πk−1 πk  πk+1 h y h y h y k−1 d k d k+1 Vh −→ Vh −→ Vh

For now, boundedness is in V -norm: kπhvkV ≤ ckvkV . But later we will need W -boundedness, which is a stronger requirement. A bounded projection is quasioptimal:

j kv − πhvkV ≤ c inf kv − wkV , v ∈ V j w∈Vh

37 / 94 First consequences from the assumptions

From the subcomplex property

k−1 d k d k+1 · · · → Vh −→ Vh −→ Vh · · · →

k k is itself a closed H-complex. (We take Wh = Vh but with the W -norm.)

∗ Therefore there is a discrete adjoint operator dh (its domain is all of k Wh ), a discrete Hodge decomposition

k k ¨ k ¨ ∗ Vh = Bh Hh Bkh.

and a discrete Poincare´ inequality

P k⊥V kzkV ≤ ch kdzk, z ∈ Zh .

38 / 94 Preservation of cohomology

Theorem Given: a closed H-complex, and a choice of f.d. subspaces satisfying

the subcomplex property and admitting a V -bdd cochain projection πh. Assume also the (very weak) approximation property

k kq − πhqk < kqk, 0 6= q ∈ H .

k k Then πh induces an isomorphism from H onto Hh. Moreover, gap H, Hh ≤ sup kq − πhqkV . q∈H kqk=1

gap(H, Hh) := max sup inf ku − vkV , sup inf ku − vkV . v∈H u∈H u∈H h v∈Hh kuk=1 kvk=1

39 / 94 Uniform Poincare´ inequality and stability

Theorem Given: a closed H-complex, and a choice of f.d. subspaces satisfying

the subcomplex property and admitting a V -bdd cochain projection πh. Then P k⊥ k kvkV ≤ c kπhkkdvkV , v ∈ Zh ∩ Vh .

Corollary (Stability and quasioptimality of the mixed method) The mixed method is stable (uniform inf-sup condition) and satisﬁes

kσ − σhkV + ku − uhkV + kp − phk

≤ C( inf kσ − τkV + inf ku − vkV + inf kp − qkV k−1 v∈V k q∈V k τ∈Vh h h

+ µ inf kPBu − vkV ), k v∈Vh where µ = µh = sup k(I − πh)rk. r∈Hk ,krk=1

40 / 94 Improved error estimates

In addition to µ = k(I − πh)PHk, deﬁne δ, η = o(1) by [∗] δ = k(I − πh)K kLin(W ,W ), η = k(I − πh)d K kLin(W ,W ).

( 2 k r+1 O(h ), r > 0, When Vh ⊃ Pr , µ = O(h ), η = O(h), δ = O(h), r = 0,

Theorem Given: an H-complex satisfying the compactness property, and a choice of f.d. subspaces satisfying the subcomplex property and

admitting a W -bdd cochain projection πh. Then

kd(σ − σh)k ≤ cE(dσ), kσ − σhk ≤ c[E(σ) + ηE(dσ)],

kd(u − uh)k ≤ c{E(du) + η[E(dσ) + E(p)]},

ku − uhk ≤ c{E(u) + η[E(du) + E(σ)] 2 +(η + δ)[E(dσ) + E(p)] + µE(PBu)}.

41 / 94 Numerical tests

− grad div u + curl rot u = f in Ω (unit square), u · n = rot u = 0 on ∂Ω (magnetic BC)

grad rot 0 → H1 −−→ H(rot) −→ L2 → 0 0 1 1 σh ∈ Vh ⊂ H , uh ∈ Vh ⊂ H(rot) k−1 hσh, τi − huh, grad τi = 0,τ ∈ Vh , k hgrad σh, vi + hrot uh, rot vi + hph, vi = hf , vi, v ∈ Vh , k huh, qi = 0, q ∈ Hh.

0 1 2 Vh Lagrange Vh R-T Vh DG

All hypotheses are met. . . 42 / 94 Numerical solution of vector Laplacian, magnetic BC

kσ − σhk rate k∇(σ − σh)k rate ku − uhk rate k rot(u − uh)k rate

2.16e-04 3.03 2.63e-02 1.98 2.14e-03 1.99 1.17e-02 1.99

2.70e-05 3.00 6.60e-03 1.99 5.37e-04 1.99 2.93e-03 2.00

3.37e-06 3.00 1.65e-03 2.00 1.34e-04 2.00 7.33e-04 2.00

4.16e-07 3.02 4.14e-04 2.00 3.36e-05 2.00 1.83e-04 2.00

3 2 2 2

43 / 94 Numerical solution of vector Laplacian, Dirichlet BC

For Dirichlet boundary conditions, σ = − div u is sought in H1, u is sought in H˚(rot) (the BC u · t = 0 is essential, u · n = 0 is natural).

There is no complex, so our theory does not apply.

kσ − σhk rate k∇(σ − σh)k rate ku − uhk rate k rot(u − uh)k rate

1.90e-02 1.62 2.53e+00 0.63 1.22e-03 2.01 1.55e-02 1.58

6.36e-03 1.58 1.68e+00 0.60 3.05e-04 2.00 5.33e-03 1.54

2.18e-03 1.54 1.14e+00 0.56 7.63e-05 2.00 1.85e-03 1.52

7.58e-04 1.52 7.89e-01 0.53 1.91e-05 2.00 6.49e-04 1.51

1.5 0.5 2 1.5 DNA-Falk-Gopalakrishnan M3AS 2011

44 / 94 Eigenvalue problems

Find λ ∈ R, 0 6= u ∈ D(L) s.t. Lu = λu, u ⊥ H

λkuk2 = kduk2 + kd∗uk2 > 0 so λ > 0 and Ku = λ−1u.

By the compactness property, K : W k → W k is compact and

self-adjoint, so0 < λ1 ≤ λ2 ≤ · · · → ∞. Denote by vi corresponding orthonormal eigenvalues, Ei = Rvi . Mixed discretization: k−1 k k Find λh ∈ R, 0 6= (σh, uh, ph) ∈ Vh × Vh × Hh s.t.

k−1 hσh, τi − huh, dτi = 0,τ ∈ Vh , k hdσh, vi + hduh, dvi + hph, vi = λhhuh, vi, v ∈ Vh , k huh, qi = 0, q ∈ Hh.

0 < λ1h ≤ λ2h ≤ ... ≤ λNhh, vih orthonormal, Eih = Rvih 45 / 94 Convergence of eigenvalue problems

Pm(j) Let i=1 Ei be the span of the eigenspaces of the ﬁrst j distinct eigenvalues. The method converges if ∀ j, > 0, ∃ h0 > 0 s.t.

m(j) m(j) ! X X max |λi −λih| ≤ and gap Ei , Ei,h ≤ if h ≤ h0. 1≤i≤m(j) i=1 i=1

A sufﬁcient (and necessary) condition for eigenvalue convergence is

operator norm convergence of the discrete solution operator KhPh to K (Kato, Babuska–Osborn, Bofﬁ–Brezzi–Gastaldi):

W →Wh orthog. The mixed discretization of the eigenvalue problem converges if

lim kKhPh − K kL(W ,W ) = 0. h→0

46 / 94 Eigenvalue convergence follows from improved estimates

2 ku−uhk ≤ c{E(u)+η[E(du)+E(σ)]+(η +δ)[E(dσ)+E(p)]+µE(PBu)}

E(dσ) + E(p) + E(PBu) ≤ kdσk + kpk + kuk ≤ kf k

E(u) ≤ δkf k, E(du) + E(σ) ≤ ηkf k

Therefore 2 k(K − KhPh)f k ≤ δ + η + µ → 0

Rates of convergence also follow, included doubled convergence rates for eigenvalues. . .

47 / 94 Exterior calculus The de Rham complex for a domain in Rn

d/dx 1-D: 0 → C∞(Ω) −−−→ C∞(Ω) → 0

∞ grad ∞ 2 rot ∞ 2-D: 0 → C (Ω) −−→ C (Ω, R ) −→ C (Ω) → 0

∞ grad ∞ 3 curl ∞ 3 div ∞ 3-D: 0 → C (Ω) −−→ C (Ω, R ) −−→ C (Ω, R ) −→ C (Ω) → 0

d d d d n-D: 0 → Λ0(Ω) −→ Λ1(Ω) −→ Λ2(Ω) −→· · · −→ Λn(Ω) → 0

k ∞ n×···×n The space Λ (Ω) = C (Ω, Rskw ), the space of smooth differential k-forms on Ω.

48 / 94 Exterior algebra

Multilinear forms on an n-dimensional vector space V k times k z }| { 0 Lin V : k-linear maps ω : V × · · · × V → R Lin V := R Lin1 V = V ∗ covectors, 1-forms tensor product: (ω ⊗ µ)(v1,..., vj+k ) = ω(v1,..., vj )µ(vj+1,..., vj+k ) dim Link V = nk

1 n 1 n i dual basis for Lin R : dx ,. . . ,dx with dx (ej ) = δij

k n σ1 σk basis for Lin R : dx ⊗ · · · ⊗ dx , 1 ≤ σ1, . . . , σk ≤ n

Alternating multilinear forms (algebraic k-forms)

k ω ∈ Alt V if ω(..., vi ,..., vj ,...) = −ω(..., vj ,..., vi ,...) k n dim Alt V = k skew part: (skw ω)(v ,..., v ) = 1 P sign(σ)ω(v ,..., v ) 1 k k! σ∈Σk σ1 σk j+k exterior product: ω ∧ µ = j skw(ω ⊗ µ)

k n σ1 σk basis for Alt R : dx ∧ · · · ∧ dx , 1 ≤ σ1 < . . . < σk ≤ n 49 / 94 Exterior algebra continued

k k−1 If ω ∈ Alt V , v ∈ V , the interior product ωyv ∈ Alt V is

ωyv(v1,..., vk−1) = ω(v, v1,..., vk−1) (ω ∧ η)yv = (ωyv) ∧ η ± ω ∧ (ηyv)

If V has an inner product, pick any orthonormal basis v1,..., vn and deﬁne P k hω, ηi = σ ω(vσ(1),..., vσ(k))η(vσ(1),..., vσ(k)), ω, η ∈ Alt V (sum over σ : {1,..., k} → {1,..., n} increasing).

n dim Alt V = 1. Fix the volume form by vol(v1,..., vn) = ±1. An orientation for V ﬁxes the sign.

∼= Hodge star: ? : Altk V −→ Altn−k V deﬁned by ω ∧ µ = h?ω, µi vol, ω ∈ Altk V , µ ∈ Altn−k V ? ? ω = ±ω

n 1 n σ σ σ∗ σ∗ On R , vol = dx ∧ ··· ∧dx = det, ? dx 1∧ ··· ∧dx k = ±dx 1 ∧ ··· ∧dx n−k Pullback: If L : V → W linear, L∗ : Altk W → Altk V is deﬁned ∗ L ω(w1,..., wk ) = ω(Lw1,..., Lwn)

50 / 94 Exterior algebra in R3

0 3 Alt R = R c ↔ c 1 3 ∼= 3 Alt R −→ R u1 dx1 + u2 dx2 + u3 dx3 ↔ u ∼ vector proxy 2 3 = 3 Alt R −→ R u1 dx2∧dx3 − u2 dx1∧dx3 + u3 dx1∧dx2 ↔ u 3 3 ∼= Alt R −→ R c ↔ c dx1∧dx2∧dx3

1 3 1 3 2 3 3 3 3 ∧ : Alt R × Alt R → Alt R × : R × R → R exterior prod. 1 3 2 3 3 3 3 3 ∧ : Alt R × Alt R → Alt R · : R × R → R

∗ 0 3 0 3 L : Alt R → Alt R id : R → R ∗ 1 3 1 3 T 3 3 L : Alt R → Alt R L : R → R pullback ∗ 2 3 2 3 3 3 L : Alt R → Alt R adj L : R → R ∗ 3 3 3 3 L : Alt R → Alt R (det L): R → R (c 7→ c det L)

1 3 0 3 3 yv : Alt R → Alt R v · : R → R 2 3 1 3 3 3 interior prod. yv : Alt R → Alt R v × : R → R 3 3 2 3 3 yv : Alt R → Alt R v : R → R (c 7→ cv)

k 3 3 inner prod. inner product on Alt R dot product on R and R

0 3 3 3 ? : Alt R → Alt R id : R → R Hodge star 1 3 2 3 3 3 ? : Alt R → Alt R id : R → R 51 / 94 Exterior calculus

k A differential k-form on a manifold M is a map x ∈ M 7→ ωx ∈ Alt Tx M. ω takes a point x ∈ M and k-tangent vectors and returns a number. 0-forms are functions, 1-forms are covector ﬁelds. We write ω ∈ Λk (M) or CΛk (M) if its continuous, C∞Λk (M) if its smooth, etc.

k If M = Ω ⊂ Rn, ω :Ω → Alt Rn. The general element of Λk (Ω) is P σ1 σk ω = σ aσ dx ∧ ··· ∧dx with aσ :Ω → R. 0 0 0 A smooth map φ : M → M , induces a linear maps φx : Tx M → Tx M and so a pullback φ∗ :Λk (M0) → Λk (M) on differential forms: ∗ 0 ∗ ∗ 0 0 (φ ω)x = φx ωφ(x) or (φ ω)x (v1,..., vk ) = ωφ(x) φx v1, . . . , φx vk ∗ ∗ ∗ φ (ω∧µ) = (φ ω)∧(φ µ) pullback of inclusion deﬁnes trace

σ σ k For ω = a dx 1 ∧ ··· ∧dx k ∈ Λ (Ω), the exterior derivative

n X X ∂aσ k σ σ k+1 dω = dx ∧dx 1 ∧ ··· ∧dx k ∈ Λ (Ω). It satisﬁes ∂x k σ k=1 ∗ ∗ d ◦ d = 0, d(ω∧µ) = (dω)∧µ ± ω∧dµ, φ (dω) = d(φ ω) We may use any coordinate chart to deﬁne d :Λk (M) → Λk+1(M). 52 / 94 Exterior calculus continued

A differential n-form on an oriented n-dim’l manifold M may be integrated Z very geometrically (w/o requiring a metric or measure): ω ∈ R, ω ∈ Λn(M) M R ∗ R n 0 M φ ω = M0 ω, ω ∈ Λ (M ) if φ preserves orientation. For ω = f (x) vol on Ω ⊂ Rn we get what notation suggests. Z Z Stokes theorem: dω = tr ω, ω ∈ Λk−1(Ω) Ω ∂Ω Combining with Leibniz, we get the integration by parts formula Z Z Z k n−k−1 dω∧η = ± ω∧dη + tr ω∧ tr η,ω ∈ Λ (Ω), η ∈ Λ (Ω) Ω Ω ∂Ω For M an oriented Riemannian manifold we have inner prod and ? on Z Z k Alt Tx M and can deﬁne: hω, ηiL2Λk = hωx , ηx i vol = ω∧ ? η Ω This allows us to rewrite the integration by parts formula Z k−1 k hdω, µi = hω, δµi + tr ω∧ tr ?µ,ω ∈ Λ , µ ∈ Λ , ∂Ω where δµ := ± ? d ? µ is the coderivative operator. 53 / 94 Vector proxies in Rn

k 0 1 n − 1 n

Λk (Ω) functions vector ﬁelds vector ﬁelds functions

Λk (∂Ω) functions tang vctr ﬂds functions 0

k k tr :Λ (Ω) → Λ (∂Ω) u|∂Ω πt u|∂Ω u|∂Ω · n 0 d :Λk (Ω) → Λk+1(Ω) grad curl div 0

δ :Λk (Ω) → Λk−1(Ω) 0 − div curl − grad

R :Λk (Ω) → u(f ) R u · t dH R u · n dH R u dH f R f 1 f n−1 f n

∗ k 0 k 0 T 0 0 φ :Λ (Ω ) → Λ (Ω) u◦φ (φx ) (u◦φ) (adj φx )(u◦φ) (det φx )(u◦φ)

dim f = k Piola transform φ :Ω → Ω0

54 / 94 L2 differential forms on a domain in Rn

Ω ⊂ Rn Lipschitz boundary

HΛk := {u ∈ L2Λk | du ∈ L2Λk+1} H∗Λk = {u ∈ L2Λk | δu ∈ L2Λk−1} = ?HΛn−k u ∈ HΛk (Ω) =⇒ tr u ∈ H−1/2Λk (∂Ω) u ∈ H∗Λk (Ω) =⇒ tr ?u ∈ H−1/2Λn−k (∂Ω) H˚Λk = {u ∈ HΛk | tr u = 0}, H˚∗Λk = {u ∈ H∗Λk | tr ?u = 0}

Theorem If we view d as an unbdd operator L2Λk → L2Λk+1 with domain HΛk , then d∗ = δ with domain H˚∗Λk . Consequently Hk = {ω ∈ L2Λk | dω = 0, δω = 0, tr ?ω = 0}.

55 / 94 L2 de Rham complex on a domain in Rn

L2 de Rham complex:

d d d 0 → HΛ0 −→ HΛ1 −→· · · −→ HΛn → 0

Dual complex:

δ δ δ 0 ← H˚∗Λ0 ←− H˚∗Λ1 ←−· · · ←− H˚∗Λn ← 0

Theorem (R. Picard ’84) For a domain (or Riemannian manifold) w/ Lipschitz boundary the compactness property holds: HΛk ∩ H˚∗Λk is compact in L2Λk .

compactness property =⇒ Fredholm =⇒ closed

56 / 94 Finite element spaces of differential forms Finite element spaces

k k Goal: deﬁne ﬁnite element spaces Λh ⊂ HΛ (Ω) satisfying the hypotheses of approximation, subcomplexes, and bounded cochain projections. A FE space is constructed by assembling three ingredients: Ciarlet ’78 A triangulation T consisting of polyhedral elements T

For each T , a space of shape functionsV (T ), typically polynomial

For each T , a set of DOFs: a set of functionals on V (T ), each associated to a face of T . These must be unisolvent, i.e., form a basis for V (T )∗.

The FE space Vh is deﬁned as functions piecewise in V (T ) with DOFs single-valued on faces. The DOFs determine (1) the interelement

continuity, and (2) a projection operator into Vh.

57 / 94 Example: HΛ0 = H1: the Lagrange ﬁnite element family

n Elements T ∈ Th are simplices in R .

Shape fns: V (T ) = Pr (T ), some r ≥ 1.

R DOFs: u 7→ f (trf u)q, q ∈ Pr−d−1(f ), f ∈ ∆(T ), d = dim f

v ∈ ∆0(T ): u 7→ u(v) R e ∈ ∆1(T ): u 7→ e(tre u)q, q ∈ Pr−2(e) R f ∈ ∆2(T ): u 7→ f (trf u)q, q ∈ Pr−3(f ) R T : u 7→ T uq, q ∈ Pr−4(T )

Theorem

The number of DOFs = dim Pr (T ) and they are unisolvent. The imposed continuity exactly forces inclusion in H1.

58 / 94 Unisolvence for Lagrange elements in n dimensions

R Shape fns: V (T ) = Pr (T ), DOFs: u 7→ f (trf u)q, q ∈ Pr−d−1(f ), d = dim f

#∆d (T ) dim Pr−d−1(fd ) dim Pr (T ) DOF count:

n X n + 1r − 1 r + n #DOF = = = dim Pr (T ). d + 1 d n d=0

Unisolvence proved by induction on dimension (n = 1 is obvious).

Suppose u ∈ Pr (T ) and all DOFs vanish. Let f be a facet of T . Note

trf u ∈ Pr (f ) the DOFs associated to f and its subfaces applied to u coincide

with the Lagrange DOFs in Pr (f ) applied to trf u

Therefore trf u vanishes by the inductive hypothesis. Thus Qn u = ( i=0 λi )p, p ∈ Pr−n−1(T ). Choose q = p in the interior DOFs to see that p = 0.

59 / 94 Polynomial differential forms

k P σ1 σk Polynomial diff. forms: Pr Λ (Ω) σ aσ dx ∧ · · · ∧dx , aσ∈Pr (Ω) k Homogeneous polynomial diff. forms: Hr Λ (Ω)

k r + n n r + n r + k dim Pr Λ = = r k r + k k k r + n − 1 n n r + n r + k dim Hr Λ = = r k n + r r + k k

(Homogeneous) polynomial de Rham subcomplex:

0 d 1 d d n 0 −→Pr Λ −→Pr−1Λ −→· · · −→Pr−nΛ −→0

0 d 1 d d n 0 −→Hr Λ −→Hr−1Λ −→· · · −→Hr−nΛ −→0

60 / 94 The Koszul complex

n n For x ∈ Ω ⊂ R , Tx Ω may be identiﬁed with R , so the identity map can be viewed as a vector ﬁeld. The Koszul differential κ :Λk → Λk−1 is the contraction with the identity: κω = ωy id. Applied to polynomials it increases degree. κ ◦ κ = 0 giving the Koszul complex:

n κ n−1 κ 0 0 −→Pr Λ −→Pr+1Λ −→· · · Pr+nΛ −→0 κ dxi = xi , κ(ω ∧ µ) = (κω) ∧ µ ± ω ∧ (κµ) k k σ1 σ Pk i σ1 σi σ κ(f dx ∧ ··· ∧dx ) = f i=1(−) dx ∧ ··· dxd ··· ∧dx 3D Koszul complex:

3 x 2 × x 1 · x 0 0 −→Pr Λ −→Pr+1Λ −−→Pr+2Λ −→ Pr+3Λ −→0

Theorem (Homotopy formula)

k (dκ + κd)ω = (r + k)ω, ω ∈ Hr Λ .

61 / 94 Proof of the homotopy formula

k k (dκ + κd)ω = (r + k)ω, ω ∈ Hr Λ , ω ∈ Hr Λ Proof by induction on k. k = 0 is Euler’s identity. k−1 i Assume true for ω ∈ Hr Λ , and verify it for ω ∧ dx .

dκ(ω ∧ dxi ) = d(κω ∧ dxi + (−1)k−1ω ∧ xi ) = d(κω) ∧ dxi + (−1)k−1(dω) ∧ xi + ω ∧ dxi .

κd(ω ∧ dxi ) = κ(dω ∧ dxi ) = κ(dω) ∧ dxi + (−1)k dω ∧ xi .

(dκ + κd)(ω ∧ dxi ) = [(dκ + κd)ω] ∧ dxi + ω ∧ dxi = (r + k)(ω ∧ dxi ).

62 / 94 Consequences of the homotopy formula

The polynomial de Rham complex is exact (except for constant 0-forms in the kernel). The Koszul complex is exact (except for constant 0-forms in the coimage). κdω = 0 =⇒ dω = 0, dκω = 0 =⇒ κω = 0 k k+1 k−1 Hr Λ = κHr−1Λ ⊕ dHr+1Λ − k k k+1 Deﬁne Pr Λ = Pr−1Λ + κHr−1Λ − 0 0 − n n k − k k Pr Λ = Pr Λ , Pr Λ = Pr−1Λ , else Pr−1Λ ( Pr Λ ( Pr Λ − k r + n r + k − 1 r k dim P Λ = = dim Pr Λ r r + k k r + k − k k − k k R(d|Pr Λ ) = R(d|Pr Λ ), N (d|Pr Λ ) = N (d|Pr−1Λ ) The complex (with constant r)

− 0 d − 1 d d − n 0 → Pr Λ −→Pr Λ −→· · · −→Pr Λ → 0 is exact (except for constant 0-forms).

63 / 94 − Complexes mixing Pr and Pr

n−1 0 On an n-D domain there are 2 complexes beginning with Pr Λ n (or ending with Pr Λ ). At each step we have two choices:

− k − k Pr Λ Pr Λ P Λk−1 or P−Λk−1 r k r k Pr−1Λ Pr−1Λ

In 3-D:

0 d − 1 d − 2 d 3 0 → Pr Λ −→Pr Λ −→Pr Λ −→Pr−1Λ → 0.

0 d − 1 d 2 d 3 0 → Pr Λ −→Pr Λ −→Pr−1Λ −→Pr−2Λ → 0,

0 d 1 d − 2 d 3 0 → Pr Λ −→Pr−1Λ −→Pr−1Λ −→Pr−2Λ → 0, 0 d 1 d 2 d 3 0 → Pr Λ −→Pr−1Λ −→Pr−2Λ −→Pr−3Λ → 0,

64 / 94 − k The Pr Λ family of simplicial FE differential forms

Given: a mesh Th of simplices T , r ≥ 1, 0 ≤ k ≤ n, we deﬁne − k Pr Λ (Th) via:

− k Shape fns: Pr Λ (T )

DOFs:Z d−k u 7→ (trf u)∧q, q ∈ Pr+k−d−1Λ (f ), f ∈ ∆(T ), d = dim f ≥ k f

Theorem − k The number of DOFs = dim Pr Λ (T ) and they are unisolvent. The imposed continuity exactly enforces inclusion in HΛk .

65 / 94 Dimension count

X k d #DOFs = #∆d (T ) dim Pr+k−d−1Λ (R ) d≥k X n + 1r + k − 1d = d + 1 d k d≥k X n + 1 r + k − 1j + k = j + k + 1 j + k j j≥0

Simplify using the identities ab aa − c X a c a + c = = b c c a − b b + j j a − b j≥0 to get r + nr + k − 1 #DOFs = = dim P−Λk r + k k r

66 / 94 − k Proof of unisolvence for Pr Λ

Lemma ˚ k R ∧ n−k If u ∈ Pr−1Λ (T ) and T u q = 0 ∀q ∈ Pr+k−n−1Λ (T ), then u = 0.

Proof: This can be proved by an explicit choice of test function. − k Proof of unisolvence: Suppose u ∈ Pr Λ (T ) and all the DOFS R ∧ d−k vanish: f (trf u) q = 0, q ∈ Pr+k−d−1Λ (f ), f ∈ ∆(T ). − k Then trf u ∈ Pr Λ (f ) and all its DOFs vanish. By induction on dimension, tr u vanishes on the boundary. So we need to show: ˚− k R ∧ n−k u ∈ Pr Λ (T ), T u q = 0 ∀q ∈ Pr+k−n−1Λ (T )= ⇒ u = 0 k In view of lemma, we just need to show u ∈ Pr−1Λ (T ). − k k By the homotopy formula, u ∈ Pr Λ , du = 0 =⇒ u ∈ Pr−1Λ . So it remains to show that du = 0. ˚ k+1 du ∈ Pr−1Λ (T ), R ∧ R ∧ n−k−1 T du p = ± T u dp = 0 ∀p ∈ Pr+k−nΛ (T ). Therefore du = 0 by the lemma (with k → k +1).

67 / 94 k The Pr Λ family of simplicial FE differential forms

Given: a mesh Th of simplices T , r ≥ 1, 0 ≤ k ≤ n, we deﬁne k Pr Λ (Th) via:

k Shape fns: Pr Λ (T )

DOFs:Z − d−k u 7→ (trf u)∧q, q ∈ Pr+k−d Λ (f ), f ∈ ∆(T ), d = dim f ≥ k f

Theorem k The number of DOFs = dim Pr Λ (T ) and they are unisolvent. The imposed continuity exactly enforces inclusion in HΛk .

68 / 94 − The Pr family in 2D

− 0 − 1 − 2 Pr Λ Pr Λ Pr Λ Lagrange Raviart-Thomas ’76 DG

r = 1

r = 2

r = 3

69 / 94 − k The Pr Λ family in 3D

− 0 − 1 − 2 − 3 Pr Λ Pr Λ Pr Λ Pr Λ Lagrange Ned´ elec´ ’80N ed´ elec´ ’80 DG

r = 1

r = 2

r = 3

70 / 94 k The Pr Λ family in 2D

0 1 2 Pr Λ Pr Λ Pr Λ Lagrange Brezzi-Douglas-Marini ’85 DG

r = 1

r = 2

r = 3

71 / 94 k The Pr Λ family in 3D

0 1 2 3 Pr Λ Pr Λ Pr Λ Pr Λ Lagrange Ned´ elec´ ’86 Ned´ elec´ ’86 DG

r = 1

r = 2

r = 3

72 / 94 − Application of the Pr and Pr families to the Hodge Laplacian

k − k The shape function spaces Pr Λ (T ) and Pr Λ (T ) combine into de Rham subcomplexes. The DOFs connect these spaces across elements to create subspaces of HΛk (Ω). k Therefore the assembled finite element spaces Pr Λ (Th) and − k n−1 Pr Λ (Th) combine into de Rham subcomplexes (in 2 ways). The DOFs of freedom determine projections from Λk (Ω) into the finite element spaces. From Stokes thm, these commute with d. Suitably modified, we obtain bounded cochain projections. Thus the abstract theory applies. We may use any two adjacent spaces in any of the complexes.     P Λk−1(T ) P−Λk (T )  r   r    d   or −→ or  − k−1   k  Pr Λ (T ) Pr−1Λ (T )

73 / 94 Rates of convergence

Rates of convergence are determined by the improved error estimates from the abstract theory. They depend on The smoothness of the data f . The amount of elliptic regularity. The degree of of complete polynomials contained in the ﬁnite element spaces.

The theory delivers the best possible results: with sufﬁciently smooth data and elliptic regularity, the rate of convergence for each of the quantities u, du, σ, dσ, and p in the L2 norm is the best possible given the degree of polynomials used for that quantity.

Eigenvalues converge as O(h2r ).

74 / 94 Historical notes

− k The P1 Λ complex is in Whitney ’57 (Bossavit ’88).

In ’76, Dodziuk and Patodi defined a finite difference approximation based on the Whitney forms to compute the eigenvalues of the Hodge Laplacian, and proved convergence. In retrospect, that method can be better viewed as a mixed finite element method. This was a step on the way to proving the Ray-Singer conjecture, completed in ’78 by W. Miller.

k The Pr Λ complex is in Sullivan ’78.

− k Hiptmair gave a uniform treatment of the Pr Λ spaces in ’99.

The uniﬁed treatment and use of the Koszul complex is in DNA-Falk-Winther ’06.

75 / 94 Bounded cochain projections

k − k The DOFs deﬁning Pr Λ (Th) and Pr Λ (Th) determine canonical k k projection operators Πh from piecewise smooth forms in HΛ onto Λh. k However, Πh is not bounded on HΛ (much less uniformly bounded k wrt h). Πh is bounded on CΛ . 2 k k If we have a smoothing operator R,h ∈ Lin(L Λ , CΛ ) such that R,h commutes with d, we can deﬁne Q,h = ΠhR,h and obtain a bounded 2 k k operator L Λ → Λh which commutes with d (as suggested by Christiansen).

However Qh will not be a projection. We correct this by using Schoberl’s¨ trick: if the ﬁnite dimensional operator

k k Q,h| k :Λ → Λ Λh h h

is invertible, then −1 πh := (Q,h| k ) Q,h, Λh

is a bounded commuting projection. It remains to get uniform bds on πh. 76 / 94 The two key estimates

For this we need two key estimates for Q,h := ΠhR,h:

For ﬁxed , Q,h is uniformly bounded: ∀ > 0 suff. small ∃ c() > 0 s.t.

sup kQ,hkLin(L2,L2) ≤ c() h

lim kI − Q,hkLin(L2,L2) = 0 uniformly in h →0

Theorem Suppose that these two estimates hold and deﬁne −1 k k − k πh := (Q,h| k ) Q,h, where Λ is either Pr Λ (Th) or P Λ (Th). Λh h r+1 k Then, for h sufﬁciently small, πh is a cochain projection onto Λh and

s s k kω − πhωk ≤ ch kωkHsΛk , ω ∈ H Λ , 0 ≤ s ≤ r + 1.

77 / 94 The smoothing operator

The simplest deﬁnition is to take R,hu to be an average over y ∈ B1 of y ∗ y (F,h) u where F,h(x) = x + hy: Z y ∗ R,hu(x) = ρ(y)[(Feh) u](x) dy B1

Needs modiﬁcation near the boundary and for non-quasiuniform meshes.

The key estimates can be proven using macroelements and scaling.

78 / 94 Bases for the spaces Bases

Since the DOFs determine a basis for the dual space of a FE space, there is a corresponding basis for the FE space. An alternative is to use the Bernstein basis fns which are given explicitly in terms of the

barycentric coordinates λi :

1 1

0 0

0 1 2 1 0 1 2 1 3 3 3 3 i j Basis of P3 dual to the nodal DOFs. Bernstein basis functions λ0λ1.

− k k For the Pr Λ and Pr Λ families in n-dimensions, there is of course again the basis determined by the DOFs. In addition, there is an explicit basis analogous to the Bernstein basis (DNA-Falk-Winter 2009). 79 / 94 Some computations with barycentric coordinates

The barycentric coordinates λ0, . . . , λn form the dual basis for − 0 P1(T ) = P1 Λ (T )

1 n n dλ ∧ ··· ∧dλ = c vol ∈ Alt T with 1 n (dλ ∧ ··· ∧dλ )(x1 − x0,..., xn − x0) 1 c = = vol(x1 − x0,..., xn − x0) n!|T |

(−1)i More generally, dλ0∧ ... ∧dcλi ∧ ... ∧dλn = vol. n!|T |

κdλi = λi − λi (0), so k X i ∧ ∧ ∧ ∧ ∧ ∧ κ(dλσ0 ··· dλσk ) = (−1) λσi dλσ0 ... [dλσi ... dλσk + ψ, i=0 k ψ ∈ P0Λ .

80 / 94 The Whitney forms

Deﬁne the Whitney form associated to the k-face f with vertices

xσ0 ,..., xσk by

k X i − k ∧ ∧ ∧ ∧ φf = (−1) λσi dλσ0 ... [dλσi ... dλσk ∈ P1 Λ i=0

vertices: λi edges: λi dλj − λj dλi triangles: λi dλj ∧dλk − λj dλi ∧dλk + λk dλi ∧dλj etc. ( Z 0, g 6= f , If f , g ∈ ∆k (T ) then trg φf = g 1/k!, g = f − k ∴ after normalization, the Whitney forms are a basis for P1 Λ dual to the DOFs.

81 / 94 Explicit geometric bases

The Bernstein basis is an explicit alternative to the Lagrange basis for the Lagrange ﬁnite elts.

α α0 αn Pr = span{ λ := λ0 ··· λn | |α| = r } α Pr (T , f ) :=span{ λ | supp α = {σ0, . . . , σk }, |α| = r }

M Pr (T ) = Pr (T , f ) f ∼= ˚ ∼ Pr (T , f ) −→ Pr (f ) = Pr−dim f −1(f ) | {z } tr z }| {

There are similar geometric bases for all k:

∼ k M k k = ˚ k ∼ − dim f −k Pr Λ (T ) = Pr Λ (T , f ), Pr Λ (T , f ) −→ Pr Λ (f ) = Pr+k−dim f Λ (f ) tr dim f ≥k ∼ − k M − k − k = ˚− k ∼ dim f −k Pr Λ (T ) = Pr Λ (T , f ), Pr Λ (T , f ) −→ Pr Λ (f ) = Pr+k−dim f −1Λ (f ) tr dim f ≥k

82 / 94 − k Basis for Pr Λ

− k To create a basis for Pr Λ (the easier case), we consider all products

α0 αn P λ0 ··· λn φf , f ∈ ∆k (T ), αi = r − 1 This form is associated the the face whose vertices are in f or for 2 which αi > 0. E.g., λ3φ[1,2] is associated with the face [1, 2, 3]. − k These span Pr Λ . However they are not linearly independent since

k X (−1)i λ φ = 0. σi [σ0···σbi ···σk ] i=0

To get a linearly independent spanning set, we impose the extra

condition that if αi 6= 0 then i ≥ σ0 (the least vertex index of f ). E.g., 2 − 2 λ1λ2φ[1,2] and λ3φ[1,2] are included in the basis for P3 Λ but λ0λ3φ[1,2] is not.

83 / 94 − 1 − 2 Example: explicit bases for Pr Λ and Pr Λ on a tet

− 1 Pr Λ (T3)

r Edge [xi , xj ] Face [xi , xj , xk ] Tet [xi , xj , xk , xl ]

1 φij

2 λi φij , λj φij λk φij , λj φik

2 2 3 {λi , λj , λi λj }φij {λi , λj , λk }λk φij , {λi , λj , λk }λj φik λk λl φij , λj λl φik , λj λk φil

− 2 Pr Λ (T3)

r Edge [xi , xj ] Face [xi , xj , xk ] Tet [xi , xj , xk , xl ]

1 φijk

2 λi φijk , λj φijk , λk φijk λl φijk , λk φijl , λj φikl

2 2 2 3 {λi , λj , λk }φijk {λi , λj , λk , λl }λl φijk

{λi λj , λi λk , λj λk }φijk {λi , λj , λk , λl }λk φijl

{λi , λj , λk , λl }λj φikl

84 / 94 Finite element differential forms on cubical meshes k M ∗ i ∗ j Shape fns: (V ∧ W ) = πSV ∧ πT W (πS : S × T → S) i+j=k

∗ ∗ DOFs: (η ∧ ρ)(πSv∧πT w) := η(v)ρ(w)

The tensor product construction

Again there are two families (only?). One results from a tensor product construction. (DNA–Bofﬁ–Bonizzoni) Suppose we have a ﬁnite element de Rham subcomplex V on an element S ⊂ Rm: d · · · → V k −→ V k+1 → · · · V k ⊂ Λk (S)

and another, W , on another element T ⊂ Rn: d · · · → W k −→ W k+1 → · · ·

The tensor-product construction produces a new complex V ∧ W , a subcomplex of the de Rham complex on S × T .

85 / 94 The tensor product construction

and another, W , on another element T ⊂ Rn: d · · · → W k −→ W k+1 → · · ·

The tensor-product construction produces a new complex V ∧ W , a subcomplex of the de Rham complex on S × T .

k M ∗ i ∗ j Shape fns: (V ∧ W ) = πSV ∧ πT W (πS : S × T → S) i+j=k

∗ ∗ DOFs: (η ∧ ρ)(πSv∧πT w) := η(v)ρ(w)

85 / 94 − k Finite element differential forms on cubes: the Qr Λ family

Start with the simple 1-D degree r ﬁnite element de Rham complex, Vr :

0 d 1 0 −→Pr Λ (I) −→Pr−1Λ (I) −→0

u(x) −→ u 0(x) dx

− k n k Take tensor product n times: Qr Λ (I ) := (Vr ∧ · · · ∧ Vr )

Qr = Pr ⊗ Pr , Pr−1 ⊗ Pr dx1 + Pr ⊗ Pr−1 dx2, Pr−1 ⊗ Pr−1 dx1 ∧ dx2

− 0 − 1 − 2 Q1 Λ Q1 Λ Q1 Λ → → Raviart-Thomas ’76

→ → → Nedelec ’80 86 / 94 The 2nd family of ﬁnite element differential forms on cubes

k n The Sr Λ (I ) family of FEDFs: (DNA–Awanou ’12) Shape fns: αi αn For a form monomial m = x1 ··· xn dxσ1 ∧ · · · ∧ dxσk , deﬁne P deg m = αi , ldeg m = #{ i | αi = 1, αi 6= {σ1, . . . , σk }}. 5 Ex: If m = x1x2x3 dx1, deg m = 7, ldeg m = 1.

k n Hr,`Λ (I ) = span of monomials with deg = r, ldeg ≥ `, k n M k+1 n Jr Λ (I ) = κHr+`−1,`Λ (I ), `≥1 k n k n k n k−1 n Sr Λ (I ) = Pr Λ (I ) ⊕ Jr Λ (I ) ⊕ dJr+1Λ (I ).

R ∧ d−k n DOFs: u 7→ f u q, q ∈ Pr−2d Λ (f ), f ∈ ∆(I )

87 / 94 Key properties

For any n ≥ 1, r ≥ 1, 0 ≤ k ≤ n:

k n k n k n Degree property: Pr Λ (I ) ⊂ Sr Λ (I ) ⊂ Pr+n−k Λ (I )

k n k n Inclusion property: Sr Λ (I ) ⊂ Sr+1Λ (I )

n k n k Trace property: For each face f of I , trf Sr Λ (I ) = Sr Λ (f ).

k n k+1 n Subcomplex property: dSr Λ (I ) ⊂ Sr−1Λ (I )

Unisolvence: The indicated DOFs are correct in number and are unisolvent.

Commuting projections: The DOFs determine commuting projections from the de Rham complex to the subcomplex

0 n d 1 n d d n n Sr Λ (I ) −→Sr−1Λ (I ) −→· · · −→Sr−nΛ (I ).

88 / 94 The case of 0-forms (H1 elements)

Deﬁne sdeg m of a monomial m to be the degree ignoring variables that enter linearly: sdeg x3yz2 = 5. For a polynomial p, sdeg p is the maximum over its monomials.

n n Sr (I ) = { p ∈ P(I ) | sdeg p ≤ r } DNA-Awanou ’10

2 2 r r 1D: Sr (I) = Pr (I), 2D: Sr (I ) = Pr (I ) + span[x y, xy ] serendipity

2 2 2 2 2 S1(I ) S2(I ) S3(I ) S4(I ) S5(I )

89 / 94 Serendipity 0-forms in more dimensions

Dimensions

n n n Pr (I ) Sr (I ) Qr (I ) n 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 2 3 6 10 15 21 4 8 12 17 23 4 9 16 25 36 3 4 10 20 35 56 8 20 32 50 74 8 27 64 125 216 4 5 15 35 70 126 16 48 80 136 216 16 81 256 625 1296

90 / 94 The 2nd cubic family in 2-D

0 1 2 S2Λ S1Λ S0Λ → →

0 1 2 S3Λ S2Λ S1Λ → →

k 2 Sr Λ (I ) k 1 2 3 4 5 0 4 8 12 17 23 1 8 14 22 32 44 2 3 6 10 15 21

91 / 94 The 2nd cubic family in 3-D

→ → →

92 / 94 Dimensions and low order cases

k 3 Sr Λ (I ) k 1 2 3 4 5 0 8 20 32 50 74 1 24 48 84 135 204 2 18 39 72 120 186 3 4 10 20 35 56

1 3 2 3 S1Λ (I ) S1Λ (I ) new element corrected element

93 / 94 The 3D shape functions in traditional FE language

0 Sr Λ : polynomials u such that sdeg u ≤ r

1 Sr Λ :

(v1, v2, v3) + (x2x3(w2 − w3), x3x1(w3 − w1), x1x2(w1 − w2)) + grad u,

vi ∈ Pr , wi ∈ Pr−1 independent of xi , sdeg u ≤ r + 1

2 Sr Λ :

(v1, v2, v3) + curl(x2x3(w2 − w3), x3x1(w3 − w1), x1x2(w1 − w2)),

3 vi , wi ∈ Pr (I ) with wi independent of xi

3 Sr Λ : v ∈ Pr

94 / 94