h(curl) and h(div) elements on polytopes from generalized

H(curl) and H(div) Elements on Polytopes fromGeneralized Barycentric Coordinates

Andrew Gillette

Department of MathematicsUniversity of Arizona

joint work with

Alex Rand (CD-adapco)Chandrajit Bajaj (UT Austin)

Andrew Gillette - U. Arizona ( )Vector Elements on Polytopes with GBCs Finite Element Circus 2014 1 / 13

The generalized barycentric coordinate approach

Let P be a convex polytope with vertex set V . We say that

λv : P → R are generalized barycentric coordinates (GBCs) on P

if they satisfy λv ≥ 0 on P and L =∑v∈V

L(vv)λv, ∀ L : P → R linear.

Familiar properties are implied by this definition:∑v∈V

λv ≡ 1︸ ︷︷ ︸partition of unity

∑v∈V

vλv(x) = x︸ ︷︷ ︸linear precision

λvi (vj ) = δij︸ ︷︷ ︸interpolation

traditional FEM family of GBC reference elements

Bilinear Map

Physical

Element

Reference

Element

Affine Map TUnit

Diameter

TΩΩ

Andrew Gillette - U. Arizona ( )Vector Elements on Polytopes with GBCs Finite Element Circus 2014 2 / 13

Developments in GBC FEM theory1 Characterization of the dependence of error estimates on polytope geometry.

vs.

2 Construction of higher order scalar-valued methods using λv functions.

λi λiλj ψij

3 Construction of H(curl) and H(div) methods using λv and ∇λv functions.

H1 grad // H(curl) curl // H(div)div // L2

λi λi∇λj λi∇λj ×∇λk χP

Andrew Gillette - U. Arizona ( )Vector Elements on Polytopes with GBCs Finite Element Circus 2014 3 / 13

Many choices of generalized barycentric coordinates

Triangulation⇒ FLOATER, HORMANN, KÓS, A generalconstruction of barycentric coordinatesover convex polygons, 2006

0 ≤ λTmi (x) ≤ λi (x) ≤ λTM

i (x) ≤ 1

Wachspress⇒ WACHSPRESS, A Rational FiniteElement Basis, 1975.⇒ WARREN, Barycentric coordinates forconvex polytopes, 1996.

Sibson / Laplace⇒ SIBSON, A vector identity for theDirichlet tessellation, 1980.⇒ HIYOSHI, SUGIHARA, Voronoi-basedinterpolation with higher continuity, 2000.

Andrew Gillette - U. Arizona ( )Vector Elements on Polytopes with GBCs Finite Element Circus 2014 4 / 13

Many choices of generalized barycentric coordinates

x

vi

vi+1

vi−1

αi−1

αi ri

∆u = 0

Mean value⇒ FLOATER, Mean value coordinates, 2003.⇒ FLOATER, KÓS, REIMERS, Mean value coordinates in3D, 2005.

Harmonic⇒ WARREN, SCHAEFER, HIRANI, DESBRUN, Barycentriccoordinates for convex sets, 2007.⇒ CHRISTIANSEN, A construction of spaces of compatibledifferential forms on cellular complexes, 2008.

Many more papers could be cited (maximum entropy coordinates, moving leastsquares coordinates, surface barycentric coordinates, etc...)

Andrew Gillette - U. Arizona ( )Vector Elements on Polytopes with GBCs Finite Element Circus 2014 5 / 13

From scalar to vector elements

The classical finite element sequences for a domain Ω ⊂ Rn are written:

n = 2 : H1 grad // H(curl) oo rot // H(div)div // L2

n = 3 : H1 grad // H(curl) curl // H(div)div // L2

These correspond to the L2 deRham diagrams from differential topology:

n = 2 : HΛ0 d0 // HΛ1 ∼= // HΛ1oo d1 // HΛ2

n = 3 : HΛ0 d0 // HΛ1 d1 // HΛ2 d2 // HΛ3

Conforming finite element subspaces of HΛk are of two types:

Pr Λk := k -forms with degree r polynomial coefficients

P−r Λk := Pr−1Λk ⊕ certain additional k -forms

This notation, from Finite Element Exterior Calculus, can be used to describe manywell-known finite element spaces.

ARNOLD, FALK, WINTHER Finite Element Exterior Calculus, Bulletin of the AMS, 2010.

Andrew Gillette - U. Arizona ( )Vector Elements on Polytopes with GBCs Finite Element Circus 2014 6 / 13

Classical finite element spaces on simplices

n=2 (triangles)

k dim space type classical description0 3 P1Λ0 H1 Lagrange elements of degree ≤ 1

3 P−1 Λ0 H1 Lagrange elements of degree ≤ 11 6 P1Λ1 H(div) Brezzi-Douglas-Marini H(div) elements of degree ≤ 1

3 P−1 Λ1 H(div) Raviart-Thomas elements of order 02 3 P1Λ2 L2 discontinuous linear

1 P−1 Λ2 L2 discontinuous piecewise constant

n=3 (tetrahedra)

0 4 P1Λ0 H1 Lagrange elements of degree ≤ 14 P−1 Λ0 H1 Lagrange elements of degree ≤ 1

1 12 P1Λ1 H(curl) Nédélec second kind H(curl) elements of degree ≤ 16 P−1 Λ1 H(curl) Nédélec first kind H(curl) elements of order 0

2 12 P1Λ2 H(div) Nédélec second kind H(div) elements of degree ≤ 14 P−1 Λ2 H(div) Nédélec first kind H(div) elements of order 0

3 4 P1Λ3 L2 discontinuous linear1 P−1 Λ3 L2 discontinuous piecewise constant

Andrew Gillette - U. Arizona ( )Vector Elements on Polytopes with GBCs Finite Element Circus 2014 7 / 13

Basis functions on simplices

n=2 (triangles)

k dim space type basis functions0 3 P1Λ0 H1 λi

1 6 P1Λ1 H(curl) λi∇λj

6 P1Λ1 H(div) rot(λi∇λj )

3 P−1 Λ1 H(curl) λi∇λj − λj∇λi

3 P−1 Λ1 H(div) rot(λi∇λj − λj∇λi )

2 3 P1Λ2 L2 piecewise linear functions1 P−1 Λ2 L2 piecewise constant functions

n=3 (tetrahedra)

0 4 P1Λ0 H1 λi

1 12 P1Λ1 H(curl) λi∇λj

6 P−1 Λ1 H(curl) λi∇λj − λj∇λi

2 12 P1Λ2 H(div) λi∇λj ×∇λk

4 P−1 Λ2 H(div) (λi∇λj ×∇λk ) + (λj∇λk ×∇λi ) + (λk∇λi ×∇λj )

3 4 P1Λ3 L2 piecewise linear functions1 P−1 Λ3 L2 piecewise constant functions

Andrew Gillette - U. Arizona ( )Vector Elements on Polytopes with GBCs Finite Element Circus 2014 8 / 13

Essential properties of basis functionsThe vector-valued basis constructions (0 < k < n) have two key properties:

1 Global continuity in H(curl) or H(div)

λi∇λj agree on tangentialcomponents at element interfaces =⇒ H(curl) continuity

λi∇λj ×∇λk agree on normalcomponents at element interfaces =⇒ H(div) continuity

2 Reproduction of requisite polynomial differential forms.

For i, j ∈ 1, 2, 3:

spanλi∇λj = span[

10

],

[01

],

[x0

],

[0x

],

[y0

],

[0y

]∼= P1Λ1(R2)

spanλi∇λj − λj∇λi = span[

10

],

[01

],

[xy

]∼= P−1 Λ1(R2)

Using generalized barycentric coordinates, we can extend all these results topolygonal and polyhedral elements.

Andrew Gillette - U. Arizona ( )Vector Elements on Polytopes with GBCs Finite Element Circus 2014 9 / 13

Basis functions on polygons and polyhedra

Theorem [G., Rand, Bajaj, 2014]Let P be a convex polygon or polyhedron. Given any set of generalized barycentriccoordinates λi associated to P, the functions listed below have global continuityand polynomial differential form reproduction properties as indicated.

n=2(polygons)

k space type functions

1 P1Λ1 H(curl) λi∇λj

P1Λ1 H(div) rot(λi∇λj )

P−1 Λ1 H(curl) λi∇λj − λj∇λi

P−1 Λ1 H(div) rot(λi∇λj − λj∇λi )

n=3(polyhedra)

1 P1Λ1 H(curl) λi∇λj

P−1 Λ1 H(curl) λi∇λj − λj∇λi

2 P1Λ2 H(div) λi∇λj ×∇λk

P−1 Λ2 H(div) (λi∇λj ×∇λk ) + (λj∇λk ×∇λi )+ (λk∇λi ×∇λj )

Note: The indices range over all pairs or triples of vertex indices from P.Andrew Gillette - U. Arizona ( )Vector Elements on Polytopes with GBCs Finite Element Circus 2014 10 / 13

Polynomial differential form reproduction identities

Let P ⊂ R3 be a convex polyhedron with vertex set vi. Let x =[x y z

]T .

Then for any 3× 3 real matrix A, ∑i,j

λi∇λj (vj − vi )T = I∑

i,j

(Avi · vj )(λi∇λj ) = Ax

12

∑i,j,k

λi∇λj ×∇λk ((vj − vi )× (vk − vi ))T = I

12

∑i,j,k

(Avi · (vj × vk ))(λi∇λj ×∇λk ) = Ax.

By appropriate choice of constant entries for A, the column vectors of I and Ax spanP1Λ1 ⊂ H(curl) or P1Λ2 ⊂ H(div).

→ Additional identities for the remaining cases are stated in:G, RAND, BAJAJ Construction of Scalar and Vector Finite Element Families onPolygonal and Polyhedral Meshes, arXiv:1405.6978, 2014

Andrew Gillette - U. Arizona ( )Vector Elements on Polytopes with GBCs Finite Element Circus 2014 11 / 13

Reducing the basisIn some cases, it should be possible to reduce the size of the basis constructed by ourmethod, in an analogous fashion to the quadratic scalar case.

n=2 (polygons)

k space # construction # boundary # polynomial

0 P1Λ0(m)/P−0 Λ1(m) v v 3

1 P1Λ1(m) v(v − 1) 2e 6

P−1 Λ1(m)

(v2

)e 3

2 P1Λ2(m)v(v − 1)(v − 2)

20 3

P−1 Λ2(m)

(v3

)0 1

→ The n = 3 (polyhedra) version of this table is given in:

G, RAND, BAJAJ Construction of Scalar and Vector Finite Element Families onPolygonal and Polyhedral Meshes, arXiv:1405.6978, 2014

Andrew Gillette - U. Arizona ( )Vector Elements on Polytopes with GBCs Finite Element Circus 2014 12 / 13

Acknowledgments

Chandrajit Bajaj UT Austin

Alexander Rand UT Austin / CD-adapco

Happy birthday, Doug!

Slides and pre-prints: http://math.arizona.edu/~agillette/

More on GBCs: http://www.inf.usi.ch/hormann/barycentric

Andrew Gillette - U. Arizona ( )Vector Elements on Polytopes with GBCs Finite Element Circus 2014 13 / 13