The Periodic Table of Finite Elements Douglas N. Arnold, University of Minnesota Collaborators: Awanou, Boffi, Bonizzoni, Falk, Winther FEniCS 2013

The Lagrange finite element spaces, Pr (Th ) u Elements: A triangulation Th consisting of simplices T u Shape functions: V (T ) = Pr (T ), some r ≥ 1 u Degrees of freedom (which must be unisolvent): v ∈ ∆0 (T ): u 7→ u (v ) e ∈ ∆1 (T ): f ∈ ∆2 (T ): T:

u 7→

R

(tre u )q , Re u 7→ f (trf u )q , R u 7→ T u q,

q ∈ Pr −2 (e) q ∈ Pr −3 (f ) q ∈ Pr −4 (T )

For a general simplex of any dimension and a face f of any dimension:

Z u 7→

(trf u )q ,

q ∈ Pr −d −1 (f ), f ∈ ∆d (T ), d ≥ 0

f

Assembled piecewise polynomials are continuous, and

Pr (Th ) = { u ∈ H 1 (Ω) | u |T ∈ V (T ) ∀T ∈ Th }

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The Maxwell eigenvalue problem with Lagrange elements Find nonzero u ∈ H (curl) such that

Z

Boffi–Gastaldi

Z curl u · curl v dx = λ

u · v dx ,

∀v ∈ H (curl)

Ω = (0, π) × (0, π), λ = m2 + n2 , m, n > 0 elts: 16

64

256

1024

4096

2.2606 4.8634 5.6530 5.6530 11.3480

2.0679 5.4030 5.4030 5.6798 9.0035

2.0171 5.1064 5.1064 5.9230 8.2715

2.0043 5.0267 5.0267 5.9807 8.0685

2.0011 5.0067 5.0067 5.9952 8.0171

1.3488 1.5349 2.4756 5.5582 5.7592

0.2576 0.4196 0.9524 1.4513 1.7446

0.0587 0.0896 0.1805 0.2938 0.3694

0.0143 0.0214 0.0417 0.0686 0.0826

0.0036 0.0053 0.0102 0.0169 0.0200

!!

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The Maxwell eigenvalue problem with H (curl) elements #V = VectorFunctionSpace(mesh, "Lagrange", 1) V = FunctionSpace(mesh, "N1curl", 1) Shape fns: (a − bx2 , c + bx1 ) elts: 16

64

DOFs: u 7→

R e

u·t

256

1024

4096

1.8577 4.1577 4.1577 8.2543 9.7268

1.9655 4.8929 4.8929 7.4306 9.8498

1.9914 4.9749 4.9749 7.8619 9.9858

1.9979 4.9938 4.9938 7.9657 9.9975

1.9995 4.9985 4.9985 7.9914 9.9994

2.1098 3.5416 4.8634 9.7268 9.7268

2.0324 4.8340 5.0962 8.0766 8.9573

2.0084 4.9640 5.0259 8.1185 9.7979

2.0021 4.9912 5.0066 8.0332 9.9506

2.0005 4.9978 5.0017 8.0085 9.9877

A good element for this problem in both theory and practice. . .

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Darcy flow u=

k

µ

grad p,

div u = f

Find (u , p) ∈ H (div) × L2 such that

Z  µ k

u · v − p div v + div u q



Z dx =

f q dx ,

∀(v , q ) ∈ H (div )×L2

Lagrange–Lagrange is singular

Lagrange–DG is unstable in > 1 dimensions

RT–DG is stable and convergent 4 / 32

Darcy flow computed with RT–DG

pressure field 5 / 32

Darcy flow computed with Lagrange–DG

pressure field 6 / 32

The Finite Element Zoo (Cubic Pavillion)

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The Finite Element Exterior Calculus Viewpoint

Differential forms and the L2 de Rham complex Differential k -forms, Λk (Ω): defined for any manifold Ω, 0 ≤ k ≤ dim Ω 0-forms are simply functions Ω → R and 1-forms are covector fields. X In local coordinates, the general k -form is X σ u= fσ dx := fσ1 ···σk dx σ1 ∧ · · · ∧ dx σk σ

1≤σ1