Duality in Finite Element Exterior Calculus and the Hodge Star - PowerPoint PPT Presentation

Duality in Finite Element Exterior Calculus and the Hodge Star Operator on the Sphere Yakov Berchenko-Kogan Washington University in St. Louis March 24, 2019 Yakov Berchenko-Kogan Duality in FEEC and the Hodge Star on the Sphere

The finite element method Figure: Degrees of freedom (blue) of piecewise linear functions (left) and piecewise quadratic functions (right). Yakov Berchenko-Kogan Duality in FEEC and the Hodge Star on the Sphere

Geometric decomposition = ⊕ 3 ⊕ 3 Figure: P 3 ( T 2 ) = ˚ P 3 ( T 2 ) ⊕ 3˚ P 3 ( T 1 ) ⊕ 3˚ P 3 ( T 0 ) ∼ (˚ 3(˚ 3(˚ ( P 3 ( T 2 )) ∗ P 3 ( T 2 )) ∗ P 3 ( T 1 )) ∗ P 3 ( T 0 )) ∗ ⊕ ⊕ = ∼ P 0 ( T 2 ) 3 P 1 ( T 1 ) 3 P 2 ( T 0 ) ⊕ ⊕ = Yakov Berchenko-Kogan Duality in FEEC and the Hodge Star on the Sphere

Duality Definition Let P and Q be spaces of functions T n → R . e.g. P = P 1 ( T 1 ), Q = ˚ P 3 ( T 1 )) Consider the pairing � ( p , q ) �→ T n pq . P and Q are dual to each other with respect to integration if this pairing is a perfect pairing P × Q → R . For each nonzero p ∈ P there exists a q ∈ Q such that � T n pq > 0, and for each nonzero q there exists such a p . Problem Construct a bijection P → Q so that for nonzero p �→ q we have q only depends on p pointwise, and � T n pq > 0. Yakov Berchenko-Kogan Duality in FEEC and the Hodge Star on the Sphere

Explicit pointwise duality y T 1 = { ( x , y ) | x + y = 1 } x Figure: Barycentric coordinates Example (Duality between P 1 ( T 1 ) and ˚ P 3 ( T 1 )) For p ∈ P 1 ( T 1 ), set q = ( xy ) p . Likewise, given q , set p = q xy . ˚ P 1 ( T 1 ) P 3 ( T 1 ) � T 1 pq x 2 y T 1 ( xy ) x 2 � x xy 2 � T 1 ( xy ) y 2 y Yakov Berchenko-Kogan Duality in FEEC and the Hodge Star on the Sphere

Finite element exterior calculus r Λ k ( T n ) of k -forms on T n with Spaces P r Λ k ( T n ) and P − polynomial coefficients of degree at most r . Special cases scalar fields Lagrange Discontinuous Galerkin vector fields Brezzi–Douglas–Marini elements Raviart–Thomas elements N´ ed´ elec elements Example P r Λ 1 ( T 3 ) and P − r Λ 1 ( T 3 ) are N´ ed´ elec H ( curl ) elements of the 2nd and 1st kinds, respectively. See Arnold, Falk, Winther, 2006. Yakov Berchenko-Kogan Duality in FEEC and the Hodge Star on the Sphere

Duality in finite element exterior calculus Theorem (Arnold, Falk, and Winther) With respect to the integration pairing � ( a , b ) �→ T n a ∧ b P r Λ k ( T n ) is dual to ˚ r + k +1 Λ n − k ( T n ) , P − r Λ k ( T n ) is dual to ˚ P r + k Λ n − k ( T n ) . P − Problem Construct an explicit bijection between these spaces so that for nonzero a �→ b we have b only depends on a pointwise, and � T n a ∧ b > 0. Yakov Berchenko-Kogan Duality in FEEC and the Hodge Star on the Sphere

A motivating example Let Ω be an 3-dimensional domain. Λ 1 (Ω) and Λ 2 (Ω) are dual to each other with respect to integration. Explicit pointwise duality Given nonzero a ∈ Λ 1 (Ω), let a = a x dx + a y dy + a z dz . Define b ∈ Λ 2 (Ω) by b = a x dy ∧ dz + a y dz ∧ dx + a z dx ∧ dy =: ∗ a . b only depends on a pointwise. . � � � � a � 2 d vol > 0 , ( a 2 x + a 2 y + a 2 a ∧ b = z ) d vol = Ω Ω Ω Yakov Berchenko-Kogan Duality in FEEC and the Hodge Star on the Sphere

The simplex and the sphere T 2 consists of points in the first orthant with x + y + z = 1. Via the change of coordinates x = u 2 , y = v 2 , z = w 2 , we obtain the unit sphere u 2 + v 2 + w 2 = 1. z w S 2 T 2 y v x u Figure: Change of coordinates Yakov Berchenko-Kogan Duality in FEEC and the Hodge Star on the Sphere

Explicit pointwise duality for finite element exterior calculus Constructing the dual 1 Start with a ∈ Λ k ( T n ). 2 Change coordinates to obtain α ∈ Λ k ( S n ). 3 Apply the Hodge star to obtain ∗ S n α ∈ Λ n − k ( S n ). 4 Multiply by the coordinate functions to obtain β ∈ Λ n − k ( S n ). in dimension 2, β = uvw ( ∗ S 2 α ). 5 Change coordinates back to obtain b ∈ Λ n − k ( T n ). Theorem (YBK) b depends on a pointwise. � T n a ∧ b > 0 for nonzero a. a ∈ P r Λ k ( T n ) iff b ∈ ˚ r + k +1 Λ n − k ( T n ) . P − r Λ k ( T n ) iff b ∈ ˚ P r + k Λ n − k ( T n ) . a ∈ P − Yakov Berchenko-Kogan Duality in FEEC and the Hodge Star on the Sphere

Example 1 Change of coordinates x = u 2 , y = v 2 , z = w 2 , dx = 2 u du , dy = 2 v dv , dz = 2 w dw . Hodge star on the sphere ν = u du + v dv + w dw , ∗ S 2 α = ∗ R 3 ( ν ∧ α ) . Example 1 a = y dy ∈ P 1 Λ 1 ( T 2 ). 2 α = 2 v 3 dv . 3 ∗ S 2 α = 2 uv 3 dw − 2 v 3 w du . 4 β = uvw ( ∗ S n α ) = 2 u 2 v 4 w dw − 2 uv 4 w 2 du . 5 b = xy 2 dz − y 2 z dx ∈ ˚ 3 Λ 1 ( T 2 ). P − Yakov Berchenko-Kogan Duality in FEEC and the Hodge Star on the Sphere

Example 2 Example 1 a = x dy − y dx ∈ P − 1 Λ 1 ( T 2 ). 2 α = 2 u 2 v dv − 2 uv 2 du . ∗ S 2 α = 2(( u 3 v + uv 3 ) dw − u 2 vw du − uv 2 w dv ) 3 . = 2 uv ( u 2 + v 2 + w 2 ) dw − uvw d ( u 2 + v 2 + w 2 ) = 2 uv dw . 4 β = uvw ( ∗ S 2 α ) = 2 u 2 v 2 w dw . 5 b = xy dz ∈ ˚ P 2 Λ 1 ( T 2 ). Integration via u -substitution � � � T 2 a ∧ b = α ∧ β = uvw ( α ∧ ∗ S 2 α ) S 2 S 2 > 0 > 0 � uvw � α � 2 d Area > 0 = S 2 > 0 Yakov Berchenko-Kogan Duality in FEEC and the Hodge Star on the Sphere

Correspondence between forms on T n and forms on S n Change of coordinates x = u 2 , y = v 2 , z = w 2 , dx = 2 u du , dy = 2 v dv , dz = 2 w dw . Definition α ∈ Λ k ( S n ) is even if it is invariant under each coordinate reflection. Let Λ k e ( S n ) denote the space of such forms. Example u 2 + v 4 w 2 uvw 2 du ∧ dv u du Theorem (YBK) The change of coordinates induces a bijection between P r Λ k ( T n ) and P 2 r + k Λ k e ( S n ) . Yakov Berchenko-Kogan Duality in FEEC and the Hodge Star on the Sphere

Even and odd forms on the sphere Definition α ∈ Λ k ( S n ) is odd if it changes sign under each coordinate reflection. Let Λ k o ( S n ) denote the space of such forms. Definition Let u N denote the product of the coordinate functions. In dimension 2, u N = uvw . Proposition If α is even, then ∗ S n α is odd, and vice versa. If α is even, then u N α is odd, and vice versa. Proof. Reflections reverse orientation, which changes the sign of ∗ S n . u N is odd. Yakov Berchenko-Kogan Duality in FEEC and the Hodge Star on the Sphere

Correspondences between forms on T n and forms on S n Theorem (YBK) Let a ∈ P r Λ k ( T n ) and α ∈ P 2 r + k Λ k e ( S n ) correspond to each other via the change of coordinates. Then for r ≥ 1 , r Λ k ( T n ) α ∈ ∗ S n P 2 r + k − 1 Λ n − k ( S n ) a ∈ P − ⇔ o a ∈ ˚ P r Λ k ( T n ) α ∈ u N P 2 r + k − n − 1 Λ k o ( S n ) ⇔ a ∈ ˚ r Λ k ( T n ) α ∈ u N ∗ S n P 2 r + k − n − 2 Λ n − k ( S n ) . P − ⇔ e Explicit pointwise duality for P r Λ k ( T n ) and ˚ r + k +1 Λ n − k ( T n ) P − 1 a ∈ P r Λ k ( T n ), 2 α ∈ P 2 r + k Λ k e ( S n ), 3 ∗ S n α ∈ ∗ S n P 2 r + k Λ k e ( S n ), 4 β = u N ∗ S n α ∈ u N ∗ S n P 2 r + k Λ k e ( S n ), 5 b ∈ ˚ r + k +1 Λ n − k ( T n ). P − Yakov Berchenko-Kogan Duality in FEEC and the Hodge Star on the Sphere

Correspondences between forms on T n and forms on S n Theorem (YBK) Let a ∈ P r Λ k ( T n ) and α ∈ P 2 r + k Λ k e ( S n ) correspond to each other via the change of coordinates. Then for r ≥ 1 , r Λ k ( T n ) α ∈ ∗ S n P 2 r + k − 1 Λ n − k ( S n ) a ∈ P − ⇔ o a ∈ ˚ P r Λ k ( T n ) α ∈ u N P 2 r + k − n − 1 Λ k o ( S n ) ⇔ a ∈ ˚ r Λ k ( T n ) α ∈ u N ∗ S n P 2 r + k − n − 2 Λ n − k ( S n ) . P − ⇔ e r Λ k ( T n ) and ˚ P r + k Λ n − k ( T n ) Explicit pointwise duality for P − 1 a ∈ P − r Λ k ( T n ), 2 α ∈ ∗ S n P 2 r + k − 1 Λ n − k ( S n ), o 3 ∗ S n α ∈ P 2 r + k − 1 Λ n − k ( S n ), o 4 β = u N ∗ S n α ∈ u N P 2 r + k − 1 Λ n − k ( S n ), o 5 b ∈ ˚ P r + k Λ n − k ( T n ). Yakov Berchenko-Kogan Duality in FEEC and the Hodge Star on the Sphere

(optional slide) Alternate characterizations of P − r Λ k ( T n ) Definition Let X be the radial vector field z X = x ∂ ∂ x + y ∂ ∂ y + z ∂ T 2 ∂ z . y x Let X T be the projection of the radial vector field to T n . � ∂ � ∂ � ∂ T 2 x − 1 y − 1 z − 1 � � � X T = ∂ x + ∂ y + ∂ z . 3 3 3 Let i X : Λ k ( T 2 ) → Λ k − 1 ( T 2 ) denote contraction. Definition (Arnold, Falk, and Winther) r Λ k ( T n ) := P r − 1 Λ k ( T n ) + i X T P r − 1 Λ k +1 ( T n ) . P − Yakov Berchenko-Kogan Duality in FEEC and the Hodge Star on the Sphere

(optional slide) Alternate characterizations of P − r Λ k ( T n ) P r Λ k ( T n ) is the restriction of P r Λ k ( R n +1 ) to T n . Likewise. . . Definition (YBK) r Λ k ( R n +1 ) := i X P r − 1 Λ k +1 ( R n +1 ) P − r Λ k ( T n ) be the restriction of P − r Λ k ( R n +1 ) to T n . Let P − Theorem (YBK) The two definitions are equivalent. Yakov Berchenko-Kogan Duality in FEEC and the Hodge Star on the Sphere

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