H(curl) and H(div) Elements on Polytopes from Generalized Barycentric Coordinates

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1 H(curl) and H(div) Elements on Polytopes from Generalized Barycentric Coordinates Andrew Gillette Department of Mathematics University 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 / 13

2 The generalized barycentric coordinate approach Let P be a convex polytope with vertex set V. We say that λ v : P R are generalized barycentric coordinates (GBCs) on P if they satisfy λ v 0 on P and L = v V L(v v)λ v L : P R linear. Familiar properties are implied by this definition: λ v 1 vλ v(x) = x v V }{{} partition of unity v V } {{ } linear precision λ vi (v j ) = δ ij }{{} interpolation traditional FEM family of GBC reference elements Bilinear Map Unit Diameter Affine Map T Reference Element Physical Element T Ω Ω Andrew Gillette - U. Arizona Vector Elements ( ) on Polytopes with GBCs Finite Element Circus / 13

3 Developments in GBC FEM theory 1 Characterization of the dependence of error estimates on polytope geometry. vs. 2 Construction of higher order scalar-valued methods using λv functions. {λi } {λi λj } {ψij } 3 Construction of H(curl) and H(div) methods using λv and λv functions. H1 grad {λi } Andrew Gillette - U. Arizona / H(curl) {λi λj } curl / H(div) {λi λj λk } Vector Elements () on Polytopes with GBCs div / L2 {χp } Finite Element Circus / 13

4 Many choices of generalized barycentric coordinates Triangulation FLOATER HORMANN KÓS A general construction of barycentric coordinates over convex polygons λ Tm i (x) λ i (x) λ T M i (x) 1 Wachspress WACHSPRESS A Rational Finite Element Basis WARREN Barycentric coordinates for convex polytopes Sibson / Laplace SIBSON A vector identity for the Dirichlet tessellation HIYOSHI SUGIHARA Voronoi-based interpolation with higher continuity Andrew Gillette - U. Arizona Vector Elements ( ) on Polytopes with GBCs Finite Element Circus / 13

5 Many choices of generalized barycentric coordinates x α i ri α i 1 v i+1 v i Mean value FLOATER Mean value coordinates FLOATER KÓS REIMERS Mean value coordinates in 3D v i 1 u = 0 Harmonic WARREN SCHAEFER HIRANI DESBRUN Barycentric coordinates for convex sets CHRISTIANSEN A construction of spaces of compatible differential forms on cellular complexes Many more papers could be cited (maximum entropy coordinates moving least squares coordinates surface barycentric coordinates etc...) Andrew Gillette - U. Arizona Vector Elements ( ) on Polytopes with GBCs Finite Element Circus / 13

6 From scalar to vector elements The classical finite element sequences for a domain Ω R n are written: n = 2 : H 1 grad n = 3 : H 1 grad H(curl) rot H(div) H(curl) curl H(div) div L 2 div L 2 These correspond to the L 2 derham diagrams from differential topology: n = 2 : HΛ 0 d 0 HΛ 1 = HΛ 1 d 1 HΛ 2 n = 3 : HΛ 0 d 0 HΛ 1 d 1 HΛ 2 d 2 HΛ 3 Conforming finite element subspaces of HΛ k are of two types: P r Λ k := k-forms with degree r polynomial coefficients P r Λ k := P r 1 Λ k {certain additional k-forms} This notation from Finite Element Exterior Calculus can be used to describe many well-known finite element spaces. ARNOLD FALK WINTHER Finite Element Exterior Calculus Bulletin of the AMS Andrew Gillette - U. Arizona Vector Elements ( ) on Polytopes with GBCs Finite Element Circus / 13

7 Classical finite element spaces on simplices n=2 (triangles) k dim space type classical description 0 3 P 1 Λ 0 H 1 Lagrange elements of degree 1 3 P 1 Λ0 H 1 Lagrange elements of degree P 1 Λ 1 H(div) Brezzi-Douglas-Marini H(div) elements of degree 1 3 P 1 Λ1 H(div) Raviart-Thomas elements of order P 1 Λ 2 L 2 discontinuous linear 1 P 1 Λ2 L 2 discontinuous piecewise constant n=3 (tetrahedra) 0 4 P 1 Λ 0 H 1 Lagrange elements of degree 1 4 P 1 Λ0 H 1 Lagrange elements of degree P 1 Λ 1 H(curl) Nédélec second kind H(curl) elements of degree 1 6 P 1 Λ1 H(curl) Nédélec first kind H(curl) elements of order P 1 Λ 2 H(div) Nédélec second kind H(div) elements of degree 1 4 P 1 Λ2 H(div) Nédélec first kind H(div) elements of order P 1 Λ 3 L 2 discontinuous linear 1 P 1 Λ3 L 2 discontinuous piecewise constant Andrew Gillette - U. Arizona Vector Elements ( ) on Polytopes with GBCs Finite Element Circus / 13

8 Basis functions on simplices n=2 (triangles) k dim space type basis functions 0 3 P 1 Λ 0 H 1 λ i 1 6 P 1 Λ 1 H(curl) λ i λ j 6 P 1 Λ 1 H(div) rot(λ i λ j ) 3 P 1 Λ1 H(curl) λ i λ j λ j λ i 3 P 1 Λ1 H(div) rot(λ i λ j λ j λ i ) 2 3 P 1 Λ 2 L 2 piecewise linear functions 1 P 1 Λ2 L 2 piecewise constant functions n=3 (tetrahedra) 0 4 P 1 Λ 0 H 1 λ i 1 12 P 1 Λ 1 H(curl) λ i λ j 6 P 1 Λ1 H(curl) λ i λ j λ j λ i 2 12 P 1 Λ 2 H(div) λ i λ j λ k 4 P 1 Λ2 H(div) (λ i λ j λ k ) + (λ j λ k λ i ) + (λ k λ i λ j ) 3 4 P 1 Λ 3 L 2 piecewise linear functions 1 P 1 Λ3 L 2 piecewise constant functions Andrew Gillette - U. Arizona Vector Elements ( ) on Polytopes with GBCs Finite Element Circus / 13

9 Essential properties of basis functions The vector-valued basis constructions (0 < k < n) have two key properties: 1 Global continuity in H(curl) or H(div) λ i λ j agree on tangential components at element interfaces λ i λ j λ k agree on normal components at element interfaces = H(curl) continuity = H(div) continuity 2 Reproduction of requisite polynomial differential forms. For i j {1 2 3}: span{λ i λ j } = span span{λ i λ j λ j λ i } = span {[ ] 1 0 {[ ] 1 0 [ ] 0 1 [ ] 0 1 [ ] x 0 [ ] 0 x [ ] y 0 [ ]} x = P 1 y Λ1 (R 2 ) [ ]} 0 = P y 1 Λ 1 (R 2 ) Using generalized barycentric coordinates we can extend all these results to polygonal and polyhedral elements. Andrew Gillette - U. Arizona Vector Elements ( ) on Polytopes with GBCs Finite Element Circus / 13

10 Basis functions on polygons and polyhedra Theorem [G. Rand Bajaj 2014] Let P be a convex polygon or polyhedron. Given any set of generalized barycentric coordinates {λ i } associated to P the functions listed below have global continuity and polynomial differential form reproduction properties as indicated. k space type functions n=2 (polygons) 1 P 1 Λ 1 H(curl) λ i λ j P 1 Λ 1 H(div) rot(λ i λ j ) P 1 Λ1 H(curl) λ i λ j λ j λ i P 1 Λ1 H(div) rot(λ i λ j λ j λ i ) n=3 (polyhedra) 1 P 1 Λ 1 H(curl) λ i λ j P 1 Λ1 H(curl) λ i λ j λ j λ i 2 P 1 Λ 2 H(div) λ i λ j λ k P 1 Λ2 H(div) (λ i λ j λ k ) + (λ j λ k λ i ) + (λ k λ i λ j ) 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 / 13

11 Polynomial differential form reproduction identities Let P R 3 be a convex polyhedron with vertex set {v i }. Let x = [ x y z ] T. Then for any 3 3 real matrix A λ i λ j (v j v i ) T = I ij (Av i v j )(λ i λ j ) = Ax ij 1 2 λ i λ j λ k ((v j v i ) (v k v i )) T = I ijk 1 2 (Av i (v j v k ))(λ i λ j λ k ) = Ax. ijk By appropriate choice of constant entries for A the column vectors of I and Ax span P 1 Λ 1 H(curl) or P 1 Λ 2 H(div). Additional identities for the remaining cases are stated in: G RAND BAJAJ Construction of Scalar and Vector Finite Element Families on Polygonal and Polyhedral Meshes arxiv: Andrew Gillette - U. Arizona Vector Elements ( ) on Polytopes with GBCs Finite Element Circus / 13

12 Reducing the basis In some cases it should be possible to reduce the size of the basis constructed by our method in an analogous fashion to the quadratic scalar case. n=2 (polygons) k space # construction # boundary # polynomial 0 P 1 Λ 0 (m)/p 0 Λ1 (m) v v 3 1 P 1 Λ 1 (m) v(v 1) 2e 6 ( ) P v 1 Λ1 (m) e P 1 Λ 2 (m) P 1 Λ2 (m) v(v 1)(v 2) 2 ( ) v 3 The n = 3 (polyhedra) version of this table is given in: G RAND BAJAJ Construction of Scalar and Vector Finite Element Families on Polygonal and Polyhedral Meshes arxiv: Andrew Gillette - U. Arizona Vector Elements ( ) on Polytopes with GBCs Finite Element Circus / 13

13 Acknowledgments Chandrajit Bajaj Alexander Rand UT Austin UT Austin / CD-adapco Happy birthday Doug! Slides and pre-prints: More on GBCs: Andrew Gillette - U. Arizona Vector Elements ( ) on Polytopes with GBCs Finite Element Circus / 13