Generalized Barycentric Coordinate Finite Element Methods on Polytope Meshes

Andrew Gillette

Department of Mathematics University of Arizona

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 1 / 41 What are a priori FEM error estimates?

n n Poisson’s equation in R : Given a domain D ⊂ R and f : D → R, find u such that strong form −∆u = f u ∈ H2(D) Z Z weak form ∇u · ∇φ = f φ ∀φ ∈ H1(D) D D Z Z 1 discrete form ∇uh · ∇φh = f φh ∀φh ∈ Vh ← finite dim. ⊂ H (D) D D Typical finite element method:

→ Mesh D by polytopes {P} with vertices {vi }; define h := max diam(P).

→ Fix basis functions λi with local piecewise support, e.g. barycentric functions. P → Define uh such that it uses the λi to approximate u, e.g. uh := i u(vi )λi

A linear system for uh can then be derived, admitting an a priori error estimate:

p p+1 ||u − uh||H1(P) ≤ Ch |u|Hp+1(P), ∀u ∈ H (P), | {z } | {z } approximation error optimal error bound provided that the λi span all degree p polynomials on each polytope P.

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 2 / 41 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 X if they satisfy λv ≥ 0 on P and L = L(vv)λv, ∀ L : P → R linear. v∈V Familiar properties are implied by this definition: X X λv ≡ 1 vλv(x) = x λvi (vj ) = δij v∈V v∈V | {z } | {z } | {z } interpolation partition of unity linear precision

traditional FEM family of GBC reference elements

Unit Affine Map T Bilinear Map Diameter

Reference Physical T Ω Ω Element Element

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 3 / 41 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.

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

{λi }{λi ∇λj }{λi ∇λj × ∇λk }{χP }

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 4 / 41 Table of Contents

1 Error estimates for linear case

2 Quadratic serendipity elements on polygons

3 Basis construction for vector-valued problems

4 Numerical results

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 5 / 41 Outline

1 Error estimates for linear case

2 Quadratic serendipity elements on polygons

3 Basis construction for vector-valued problems

4 Numerical results

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 6 / 41 Anatomy of an error estimate

In the case of functions λv associated to vertices of a polygonal mesh, we have:

X u − u( )λ ≤ C C diam(P) |u| v v proj-λ proj-linear H2(P) v H1(P) | {z } | {z } | {z } constants 2nd order approximation error oscillation in value and derivative in u P Cproj-λ ≈ operator norm of projection u 7−→ v u(v)λv

Cproj-linear ≈ operator norm of projection u 7−→ linear polynomials on P (from Bramble-Hilbert Lemma) diam(P) = diameter of polygon P.

Key question for polygonal finite element methods

What geometrical properties of P can cause Cproj-λ to be large?

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 7 / 41 Mathematical characterization

Problem statement Given a simple convex d-dimensional polytope P, define X Λ := sup |∇λv(x)| x∈P v∈V where λv are generalized barycentric coordinates on P. Find upper and lower bounds on Λ in terms of geometrical properties of P.

Remark: It can be shown that

Cproj-λ = 1 + CS(1 +Λ) where CS is the Sobolev embedding constant satisfying ||u||C0(P) ≤ CS ||u||Hk (P) independent of u ∈ Hk (P), provided that k > d/2.

Hence, bounds on Λ help us characterize when Cproj-λ is large.

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 8 / 41 The triangular case X Λ := sup |∇λv(x)| x∈P v∈V

If P is a triangle, Λ can be large when P has a large interior angle.

→ This is often called the maximum angle condition for finite elements.

Figure from:S HEWCHUK What is a good linear element? Int’l Meshing Roundtable, 2002.

BABUŠKA,AZIZ On the angle condition in the finite element method, SIAM J. Num. An., 1976.

JAMET Estimations d’erreur pour des éléments finis droits presque dégénérés, ESAIM:M2AN, 1976.

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 9 / 41 Motivation

Observe that on triangles of fixed diameter:

|∇λv| large ⇐⇒ interior angle at v is large ⇐⇒ the altitude “at v” is small

For Wachspress coordinates, we generalize to polygons:

|∇λv| large ⇐⇒ the “altitude” at v is small

and then to simple polytopes. (A simple d-dimensional polytope has exactly d faces at each vertex)

Given a simple convex d-dimensional polytope P, let

h∗ := minimum distance from a vertex to a hyper-plane of a non-incident face. X Then sup |∇λv(x)| =:Λ is large ⇐⇒ h∗ is small x∈P v∈V

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 10 / 41 Upper bound for simple convex polytopes

Theorem [Floater, G., Sukumar] d Let P be a simple convex polytope in R and let λv be generalized Wachspress 2d coordinates. Then Λ ≤ where h∗ = min min dist(v, f ) h∗ f v6∈f

normal to face f , v nf nf2 p (x) := = scaled by the reciprocal f h (x) f of the distance from x to f nf1

wv(x) := det(p (x), ··· , p (x)) h (x) hf1(x) f1 fd f2 P x volume formed by the d vectors {p (x)} = fi for the d faces incident to v

The generalized Wachspress coordinates are defined by

wv(x) λv(x) := X wu(x) u

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 11 / 41 Proof sketch for upper bound

X 2d To prove: sup |∇λv(x)| =:Λ ≤ where h∗ := min min hf (v). x h∗ f v6∈f v

1 Bound |∇λv| by summations over faces incident and not incident to v. ! ! X 1 X X 1 X |∇λv| ≤ λv 1 − λu + λv λu hf hf f ∈Fv u∈f f 6∈Fv u∈f

2 Summing over v gives a constant bound. ! ! X X 1 X X |∇λv| ≤ 2 1 − λu λu hf v f ∈F u∈f u∈f

3 Write hf (x) using λv (possible since hf is linear) and derive the bound. ! X X 1 X 1 2d Λ ≤ 2 λu = 2 |{f : f 3 v}|λv = h∗ h∗ h∗ f ∈F u∈f v∈V

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 12 / 41 Lower bound for polytopes

Theorem [Floater, G., Sukumar] d Let P be a simple convex polytope in R and let λv be any generalized barycentric coordinates on P. Then 1 ≤ Λ h∗

Proof sketch:

1 Show that h∗ = hf (w), for some particular face f of P and vertex w 6∈ f . 2 Let v be the vertex in f closest to w. Show that v f 1 |∇λw(v)| = hf (w)

w 3 Conclude the result, since 1 1 Λ ≥ |∇λw(v)| = = hf (w) h∗

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 13 / 41 Upper and lower bounds on polytopes

d X For a polytope P ⊂ R , define Λ := sup |∇λv(x)|. x∈P v

d 1 2d simple convex polytope in R ≤ Λ ≤ h∗ h∗

d 1 d + 1 d-simplex in R ≤ Λ ≤ h∗ h∗

√ d 1 d + d hyper-rectangle in R ≤ Λ ≤ h∗ h∗

2 2(1 + cos(π/k)) 4 regular k-gon in R ≤ Λ ≤ h∗ h∗

2 Note that lim 2(1 + cos(π/k)) = 4, so the bound is sharp in R . k→∞

FLOATER,G,SUKUMAR Gradient bounds for Wachspress coordinates on polytopes, SIAM J. Numerical Analysis, 2014.

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 14 / 41 Many other barycentric coordinates are available ...

Triangulation ⇒ FLOATER,HORMANN,KÓS, A general construction of barycentric coordinates over convex polygons, 2006

Tm TM 0 ≤ λi (x) ≤ λi (x) ≤ λi (x) ≤ 1

Wachspress ⇒ WACHSPRESS, A Rational Finite Element Basis, 1975. ⇒ WARREN, Barycentric coordinates for convex polytopes, 1996.

Sibson / Laplace ⇒ SIBSON, A vector identity for the Dirichlet tessellation, 1980. ⇒ HIYOSHI,SUGIHARA, Voronoi-based interpolation with higher continuity, 2000.

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 15 / 41 Many other barycentric coordinates are available ...

vi+1 Mean value ⇒ FLOATER, Mean value coordinates, 2003.

αi ⇒ FLOATER,KÓS,REIMERS, Mean value coordinates in x ri

vi 3D, 2005. αi 1 −

vi 1 − Harmonic ⇒ WARREN,SCHAEFER,HIRANI,DESBRUN, Barycentric coordinates for convex sets, 2007. ∆u = 0 ⇒ CHRISTIANSEN, A construction of spaces of compatible differential forms on cellular complexes, 2008.

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

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 16 / 41 Geometric criteria for convergence estimates

For other types of coordinates (on polygons only) we consider additional geometric measures.

Let ρ(Ω) denote the radius of the largest inscribed circle. The aspect ratio γ is defined by diam(Ω) γ = ∈ (2, ∞) ρ(Ω)

Three possible geometric conditions on a polygonal mesh:

G1. BOUNDED ASPECT RATIO: ∃ γ∗ < ∞ such that γ < γ∗

G2. MINIMUMEDGELENGTH: ∃ d∗ > 0 such that |vi − vi−1| > d∗ ∗ ∗ G3. MAXIMUMINTERIORANGLE: ∃ β < π such that βi < β

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 17 / 41 Polygonal Finite Element Optimal Convergence

Theorem [G, Rand, Bajaj] In the table, any necessary geometric criteria to achieve the a priori linear error estimate are denoted by N. The set of geometric criteria denoted by S in each row taken together are sufficient to guarantee the estimate. G1 G2 G3 (aspect (min edge (max interior ratio) length) angle)

Triangulated λTri - - S,N

Wachspress λWach S S S,N

Sibson λSibs SS-

Mean Value λMV SS-

Harmonic λHar S--

G,RAND,BAJAJ Error Estimates for Generalized Barycentric Interpolation Advances in Computational Mathematics, 37:3, 417-439, 2012 RAND,G,BAJAJ Interpolation Error Estimates for Mean Value Coordinates, Advances in Computational Mathematics, 39:2, 327-347, 2013. Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 18 / 41 Outline

1 Error estimates for linear case

2 Quadratic serendipity elements on polygons

3 Basis construction for vector-valued problems

4 Numerical results

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 19 / 41 From linear to quadratic elements

A naïve quadratic element is formed by products of linear GBCs:

pairwise {λi } / {λaλb} products

Why is this naïve? k
For a k-gon, this construction gives k + 2 basis functions λaλb The space of quadratic polynomials is only dimension 6: {1, x, y, xy, x 2, y 2} Conforming to a linear function on the boundary requires 2 degrees of freedom per edge ⇒ only 2k functions needed!

Problem Statement Construct 2k basis functions associated to the vertices and edge midpoints of an arbitrary k-gon such that a quadratic convergence estimate is obtained.

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 20 / 41 Polygonal Quadratic Serendipity Elements

We define matrices A and B to reduce the naïve quadratic basis. filled dot = Interpolatory domain point = all functions in the set evaluate to 0 except the associated function which evaluates to 1 open dot = non-interpolatory domain point = partition of unity satisfied, but not a nodal basis

pairwise A B {λi } / {λaλb} / {ξij } / {ψij } products k
k k + 2 2k 2k Linear Quadratic Serendipity Lagrange

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 21 / 41 From quadratic to serendipity

The bases are ordered as follows:

ξii and λaλa = basis functions associated with vertices ξi(i+1) and λaλa+1 = basis functions associated with edge midpoints λaλb = basis functions associated with interior diagonals, i.e. b ∈/ {a − 1, a, a + 1}

Serendipity basis functions ξij are a linear combination of pairwise products λaλb:

 11 11 11   λaλa  c11 ··· cab ··· c(n−2)n  λaλa    .  ......  . ξii  .   . . . . .   .   .     .   .   λ λ   cij ··· cij ··· cij   λ λ   .  = A  a a+1  =  11 ab (n−2)n   a a+1   .     .  ξ  .   ......   .  i(i+1)  .   . . . . .   .  n(n+1) n(n+1) n(n+1) λaλb λaλb c11 ··· cab ··· c(n−2)n

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 22 / 41 From quadratic to serendipity

We require the serendipity basis to have quadratic approximation power: X Constant precision: 1 = ξii + 2ξi(i+1) i X Linear precision: x = vi ξii + 2vi(i+1)ξi(i+1) i T X T T T Quadratic precision: xx = vi vi ξii + (vi vi+1 + vi+1vi )ξi(i+1) i Theorem [Rand, G, Bajaj]

ab ξbb Constants {cij } exist for any convex polygon such that ξb(b+1) the resulting basis {ξij } satisfies constant, linear, and

ξb(b 1) quadratic precision requirements. −

λaλb Proof: We produce a coefficient matrix A with the ξa(a 1) structure −  0  A := I A ξaa 0 ξa(a+1) where A has only six non-zero entries per column and show that the resulting functions satisfy the six precision equations.

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 23 / 41 Pairwise products vs. Lagrange basis

Even in 1D, pairwise products of barycentric functions do not form a Lagrange basis at interior degrees of freedom:

1 λ0λ0 λ1λ1 1 ψ ψ 00 ψ01 11

λ0λ1

1 1 Pairwise products Lagrange basis

Translation between these two bases is straightforward and generalizes to the higher dimensional case.

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 24 / 41 From serendipity to Lagrange

B {ξij } / {ψij }  1 −1 · · · −1   1 −1 −1 ···  ψ    ξ  11  ......  11 ψ  . . .  ξ  22    22  .     .   .   ......   .   .   . . .   .        ψnn  1 −1 −1  ξnn [ψij ] =   =     = B[ξij ]. ψ12  4  ξ12       ψ23  4  ξ23        .   .   .   .   .   .   .   0 .   .    ψn1  .  ξn1  ..  4

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 25 / 41 Serendipity Theorem

pairwise A B {λi } / {λaλb} / {ξij } / {ψij } products Theorem [Rand, G, Bajaj] Given bounds on polygonal geometric quality: ||A|| is uniformly bounded, ||B|| is uniformly bounded, and 2 span{ψij } ⊃ P2(R ) = quadratic polynomials in x and y

2 We obtain the quadratic a priori error estimate: ||u − uh||H1(Ω) ≤ Ch |u|H3(Ω)

RAND,G,BAJAJ Quadratic Serendipity Finite Element on Polygons Using Generalized Barycentric Coordinates, Math. Comp., 2011

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 26 / 41 Special case of a square

v4 v34 v3 Bilinear functions are barycentric coordinates: λ1 = (1 − x)(1 − y) λ2 = x(1 − y) v v 14 23 λ3 = xy λ4 = (1 − x)y  0  v1 v12 v2 Compute [ξij ] := I A [λaλb]         ξ11 1 0 −1 0 λ1λ1 (1 − x)(1 − y)(1 − x − y)  ξ22   0 0 0 −1   λ2λ2   x(1 − y)(x − y)           ξ33   0 0 −1 0   λ3λ3   xy(−1 + x + y)           ξ44   0 ··· 0 0 −1   λ4λ4   (1 − x)y(y − x)    =     =    ξ12   0 ··· 0 1/2 1/2   λ1λ2   (1 − x)x(1 − y)           ξ23   0 0 1/2 1/2   λ2λ3   x(1 − y)y           ξ34   0 0 1/2 1/2   λ3λ4   (1 − x)xy  ξ14 0 1 1/2 1/2 λ1λ4 (1 − x)(1 − y)y

 n 2 2 2 2o 2 span ξii , ξi(i+1) = span 1, x, y, x , y , xy, x y, xy =: S2(I )

2 Hence, this provides a computational basis for the serendipity space S2(I ) defined in ARNOLD,AWANOU The serendipity family of finite elements, Found. Comp. Math, 2011.

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 27 / 41 Outline

1 Error estimates for linear case

2 Quadratic serendipity elements on polygons

3 Basis construction for vector-valued problems

4 Numerical results

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 28 / 41 From scalar to vector elements

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

grad rot div n = 2 : H1 / H(curl) o / H(div) / L2 grad curl div n = 3 : H1 / H(curl) / H(div) / L2

These correspond to the L2 deRham diagrams from differential topology:

d0 ∼= d1 n = 2 : HΛ0 / HΛ1 o / HΛ1 / HΛ2 d d d n = 3 : HΛ0 0 / HΛ1 1 / HΛ2 2 / HΛ3

Conforming finite element subspaces of HΛk are of two types:

k Pr Λ := k-forms with degree r polynomial coefficients − k k Pr Λ := Pr−1Λ ⊕ {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, 2010.

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 29 / 41 Classical finite element spaces on simplices

n=2 (triangles) k dim space type classical description 0 1 0 3 P1Λ H Lagrange elements of degree ≤ 1 − 0 1 3 P1 Λ H Lagrange elements of degree ≤ 1 1 1 6 P1Λ H(div) Brezzi-Douglas-Marini H(div) elements of degree ≤ 1 − 1 3 P1 Λ H(div) Raviart-Thomas elements of order 0 2 2 2 3 P1Λ L discontinuous linear − 2 2 1 P1 Λ L discontinuous piecewise constant

n=3 (tetrahedra)

0 1 0 4 P1Λ H Lagrange elements of degree ≤ 1 − 0 1 4 P1 Λ H Lagrange elements of degree ≤ 1 1 1 12 P1Λ H(curl) Nédélec second kind H(curl) elements of degree ≤ 1 − 1 6 P1 Λ H(curl) Nédélec first kind H(curl) elements of order 0 2 2 12 P1Λ H(div) Nédélec second kind H(div) elements of degree ≤ 1 − 2 4 P1 Λ H(div) Nédélec first kind H(div) elements of order 0 3 2 3 4 P1Λ L discontinuous linear − 3 2 1 P1 Λ L discontinuous piecewise constant

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 30 / 41 Basis functions on simplices

n=2 (triangles) k dim space type basis functions 0 1 0 3 P1Λ H λi 1 1 6 P1Λ H(curl) λi ∇λj 1 6 P1Λ H(div) rot(λi ∇λj ) − 1 3 P1 Λ H(curl) λi ∇λj − λj ∇λi − 1 3 P1 Λ H(div) rot(λi ∇λj − λj ∇λi ) 2 2 2 3 P1Λ L piecewise linear functions − 2 2 1 P1 Λ L piecewise constant functions

n=3 (tetrahedra)

0 1 0 4 P1Λ H λi 1 1 12 P1Λ H(curl) λi ∇λj − 1 6 P1 Λ H(curl) λi ∇λj − λj ∇λi 2 2 12 P1Λ H(div) λi ∇λj × ∇λk − 2 4 P1 Λ H(div) (λi ∇λj × ∇λk ) + (λj ∇λk × ∇λi ) + (λk ∇λi × ∇λj ) 3 2 3 4 P1Λ L piecewise linear functions − 3 2 1 P1 Λ L piecewise constant functions

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 31 / 41 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 ∇λ agree on tangential i j =⇒ H(curl) continuity components at element interfaces λ ∇λ × ∇λ agree on normal i j k =⇒ H(div) continuity components at element interfaces

2 Reproduction of requisite polynomial differential forms.

For i, j ∈ {1, 2, 3}: 1 0 x 0 y 0 span{λ ∇λ } = span , , , , , =∼ P Λ1( 2) i j 0 1 0 x 0 y 1 R

1 0 x span{λ ∇λ − λ ∇λ } = span , , =∼ P−Λ1( 2) i j j i 0 1 y 1 R

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

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 32 / 41 Basis functions on polygons and polyhedra

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

k space type functions

1 1 P1Λ H(curl) λi ∇λj n=2 1 P1Λ H(div) rot(λi ∇λj ) (polygons) − 1 P1 Λ H(curl) λi ∇λj − λj ∇λi − 1 P1 Λ H(div) rot(λi ∇λj − λj ∇λi )

1 1 P1Λ H(curl) λi ∇λj − 1 P1 Λ H(curl) λi ∇λj − λj ∇λi n=3 2 (polyhedra) 2 P1Λ H(div) λi ∇λj × ∇λk − 2 P1 Λ 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. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 33 / 41 Polynomial differential form reproduction identities

3  T Let P ⊂ R be a convex polyhedron with vertex set {vi }. Let x = x y z . Then for any 3 × 3 real matrix A,

X T λi ∇λj (vj − vi ) = I i,j X (Avi · vj )(λi ∇λj ) = Ax i,j

1 X λ ∇λ × ∇λ ((v − v ) × (v − v ))T = 2 i j k j i k i I i,j,k 1 X ( v · (v × v ))(λ ∇λ × ∇λ ) = x. 2 A i j k i j k A i,j,k

By appropriate choice of constant entries for A, the column vectors of I and Ax span 1 2 P1Λ ⊂ H(curl) or P1Λ ⊂ 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:1405.6978, 2014

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 34 / 41 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 − 1 0 P1Λ (m)/P0 Λ (m) v v 3 1 1 P1Λ (m) v(v − 1) 2e 6 ! v P−Λ1(m) e 3 1 2 v(v − 1)(v − 2) 2 P Λ2(m) 0 3 1 2 ! v P−Λ2(m) 0 1 1 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:1405.6978, 2014

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 35 / 41 Outline

1 Error estimates for linear case

2 Quadratic serendipity elements on polygons

3 Basis construction for vector-valued problems

4 Numerical results

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 36 / 41 Matlab code for Wachspress coordinates on polygons

Input: The vertices v1,..., vn of a polygon and a point x

Output: Wachspress functions λi and their gradients ∇λi

function [phi dphi] = wachspress2d(v,x) n = size(v,1); w = zeros(n,1); R = zeros(n,2); phi = zeros(n,1); dphi = zeros(n,2);

un = getNormals(v); % computes the outward unit normal to each edge

p = zeros(n,2); for i = 1:n h = dot(v(i,:) - x,un(i,:)); p(i,:) = un(i,:) / h; end for i = 1:n im1 = mod(i-2,n) + 1; w(i) = det([p(im1,:);p(i,:)]); Matlab code for polygons and polyhedra R(i,:) = p(im1,:) + p(i,:); end (simple or non-simple) included in appendix of wsum = sum(w); FLOATER,G,SUKUMAR Gradient bounds for phi = w/wsum; Wachspress coordinates on polytopes, phiR = phi’ * R; SIAM J. Numerical Analysis, 2014. for k = 1:2 dphi(:,k) = phi .* (R(:,k) - phiR(:,k)); end

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 37 / 41 Numerical results

→ We fix a sequence of polyhedral meshes where h denotes the maximum diameter of a mesh element. → ∃ γ > 0 such that if any element from any mesh in the sequence is scaled to have diameter 1, the computed value of h∗ will be ≥ γ. h= 0.7071 h= 0.3955

→ We solve the weak form of the Poisson problem: Z Z 1 ∇u · ∇w dx = f w dx, ∀w ∈ H0 (Ω), Ω Ω where f (x) is defined so that the exact solution is u(x) = xyz(1 − x)(1 − y)(1 − z).

→ Using Wachspress coordinates λv, the local stiffness matrix has entries of the form Z ∇λv · ∇λw dx, which we integrate by tetrahedralizing P and using a second-order P accurate quadrature rule (4 points per tetrahedron).

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 38 / 41 Numerical results

h= 0.7071 h= 0.3955 h= 0.1977 h= 0.0989

As expected, we observe optimal convergence convergence rates: quadratic in L2 norm and linear in H1 semi-norm.

||u − uh|| |u − uh| Mesh # of nodes h 0,P Rate 1,P Rate ||u||0,P |u|1,P a 78 0.7071 2.0 × 10−1 – 4.1 × 10−1 – b 380 0.3955 5.4 × 10−2 2.28 2.1 × 10−1 1.14 c 2340 0.1977 1.4 × 10−2 1.96 1.1 × 10−1 0.97 d 16388 0.0989 3.5 × 10−3 1.99 5.4 × 10−2 0.99 e 122628 0.0494 8.8 × 10−4 2.00 2.7 × 10−2 0.99

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 39 / 41 Numerical evidence for non-affine image of a square

Instead of mapping , use quadratic serendipity GBC interpolation with mean value coordinates: n X  vi + vi+1  u = I u := u(v )ψ + u ψ h q i ii 2 i(i+1) n = 2 n = 4 i=1

Non-affine bilinear mapping Quadratic serendipity GBC method || − || ||∇( − )|| ||u − uh||L2 ||∇(u − uh)||L2 u uh L2 u uh L2 n error rate error rate n error rate error rate 2 5.0e-2 6.2e-1 2 2.34e-3 2.22e-2 4 6.7e-3 2.9 1.8e-1 1.8 4 3.03e-4 2.95 6.10e-3 1.87 8 9.7e-4 2.8 5.9e-2 1.6 8 3.87e-5 2.97 1.59e-3 1.94 16 1.6e-4 2.6 2.3e-2 1.4 16 4.88e-6 2.99 4.04e-4 1.97 32 3.3e-5 2.3 1.0e-2 1.2 32 6.13e-7 3.00 1.02e-4 1.99 64 7.4e-6 2.1 4.96e-3 1.1 64 7.67e-8 3.00 2.56e-5 1.99 128 9.59e-9 3.00 6.40e-6 2.00 ARNOLD,BOFFI,FALK, Math. Comp., 2002 256 1.20e-9 3.00 1.64e-6 1.96

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 40 / 41 Acknowledgments

Chandrajit Bajaj UT Austin Alexander Rand UT Austin / CD-adapco Michael Holst UC San Diego Michael Floater University of Oslo N. Sukumar UC Davis

Thanks for the invitation to speak!

Slides and pre-prints: http://math.arizona.edu/~agillette/ More on GBCs: http://www.inf.usi.ch/hormann/barycentric

Andrew Gillette - U. ArizonaGBC FEM( ) on Polytope Meshes CERMICS - June 2014 41 / 41