Analysis and Applications of Polygonal and Serendipity Finite Element Methods

Analysis and Applications of Polygonal and Serendipity Finite Element Methods Andrew Gillette Department of Mathematics University of California, San Diego http://ccom.ucsd.edu/ agillette/ Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 1 / 45

What are a priori FEM error estimates? Poisson s equation in 2D: Given a domain D R 2 and f : D R, find u such that strong form u = f u H 2 (D) weak form u φ = f φ φ H 1 (D) discrete form D D u h φ h = Typical finite element method: D D f φ h φ h V h finite dimen. H 1 (D) Mesh D by polygons {Ω} with vertices {v i }; define h := max diam(ω). Fix basis functions λ i with local piecewise support, e.g. barycentric functions. Define u h such that it uses the λ i to approximate u, e.g. u h := i u(v i)λ i A linear system for u h can then be derived, admitting an a priori error estimate: u u h H 1 (Ω) C h p u H }{{} 2 (Ω), u H p+1 (Ω), }{{} approximation error optimal error bound provided that the λ i span all degree p polynomials on each polygon Ω. Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 2 / 45

Modern applications require new theory u u h H 1 (Ω) C h p u H 2 (Ω), u H2 (Ω) Modern applications such as patient-specific cardiac electrophysiology need efficient, stable, error-bounded real-time methods. These methods require both accurate geometry approximation (small h) and accurate function approximation (large p), but both are computationally expensive. Two trends in Finite Element Methods research can help: 1 Polygonal & polyhedral generalized barycentric coordinate FEM Greater geometric flexibility from polygonal meshing can alleviate known difficulties with simplicial and cubical elements. 2 Serendipity finite element methods Long observed but only recently formalized theory ensuring order p function approximation with many fewer basis functions than expected. Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 3 / 45

Table of Contents 1 Linear Polygonal Elements with GBCs 2 Quadratic Serendipity Elements on Polygons 3 Cubic Hermite Serendipity Elements on Cubes 4 Current and Future Work Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 4 / 45

Outline 1 Linear Polygonal Elements with GBCs 2 Quadratic Serendipity Elements on Polygons 3 Cubic Hermite Serendipity Elements on Cubes 4 Current and Future Work Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 5 / 45

The generalized barycentric coordinate approach Fix Ω R 2 a convex polygon with vertices v 1,..., v n. We call functions λ i : Ω R generalized barycentric coordinates (GBCs) if they satisfy: λ i 0 on Ω L = n L(v i )λ i, i=1 L : Ω R linear Familiar properties are implied by this definition: n n λ i 1 v i λ i (x) = x i=1 }{{} partition of unity i=1 } {{ } linear precision λ i (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 - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 6 / 45

Many generalizations to choose from... Triangulation FLOATER, HORMANN, KÓS, A general construction of barycentric coordinates over convex polygons, 2006 0 λ Tm i (x) λ i (x) λ T M i (x) 1 Wachspress WACHSPRESS, A Rational Finite Element Basis, 1975. Sibson / Laplace SIBSON, A vector identity for the Dirichlet tessellation, 1980. HIYOSHI, SUGIHARA, Voronoi-based interpolation with higher continuity, 2000. Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 7 / 45

Many generalizations to choose from... x α i ri v i+1 Mean value FLOATER, Mean value coordinates, 2003. FLOATER, KÓS, REIMERS, Mean value coordinates in 3D, 2005. α i 1 v i v i 1 u = 0 Harmonic WARREN, Barycentric coordinates for convex polytopes, 1996. WARREN, SCHAEFER, HIRANI, DESBRUN, Barycentric coordinates for convex sets, 2007. 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 - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 8 / 45

Comparison via eyeball norm Andrew Gillette - UCSD Triangulated Triangulated Wachspress Mean Value Polygonal () & Serendipity FEM SMU - Feb 2013 9 / 45

Comparison via eyeball norm Wachspress Sibson Mean Value Discrete Harmonic Andrew Gillette - UCSD Polygonal () & Serendipity FEM SMU - Feb 2013 10 / 45

Optimal Convergence Estimates on Polygons Let Ω be a convex polygon with vertices v 1,..., v n. For linear elements, an optimal convergence estimate has the form n u u(v i )λ i C diam(ω) u H 2 (Ω), u H 2 (Ω). (1) i=1 H }{{ 1(Ω) }{{}} optimal error bound approximation error The Bramble-Hilbert lemma in this context says that any u H 2 (Ω) is close to a first order polynomial in H 1 norm. VERFÜRTH, A note on polynomial approximation in Sobolev spaces, Math. Mod. Num. An., 2008. DEKEL, LEVIATAN, The Bramble-Hilbert lemma for convex domains, SIAM J. Math. An., 2004. For (1), it suffices to prove an H 1 -interpolant estimate over domains of diameter one: n u(v i )λ i C I u H 2 (Ω), u H1 (Ω). (2) H 1(Ω) i=1 For (2), it suffices to bound the gradients of the {λ i }, i.e. prove C λ R such that λ i L 2 (Ω) C λ. (3) Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 11 / 45

Geometric Hypotheses for Convergence Estimates v1 v1 To bound the gradients of the coordinates, we need control of the element geometry. Let ρ(ω) denote the radius of the largest inscribed circle. The aspect ratio γ is defined by γ = diam(ω) ρ(ω) (2, ) Three possible geometric conditions on a polygonal mesh: G1. BOUNDED ASPECT RATIO: γ < such that γ < γ G2. MINIMUM EDGE LENGTH: d > 0 such that v i v i 1 > d G3. MAXIMUM INTERIOR ANGLE: β < π such that β i < β Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 12 / 45

A key geometric proposition Proposition Suppose G1 (max aspect ratio γ ) and G2 (min edge length d ) hold. Define h := d 2γ (1 + d ) > 0 Then for all x Ω, B(x, h ) does not intersect any two non-adjacent edges of Ω. Proof: Suppose B(x, h) intersects two non-adjacent edges for some h > 0. The wedge W formed by these edges can be used to show that either G1 fails (contradiction) or h > h. Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 13 / 45

Gradient bound for mean value coordinates x αi αi 1 vi 1 ri vi+1 Theorem vi The mean value coordinates are defined by λ MV w i (x) i (x) := n j=1 w j(x) tan w i (x) := ( αi (x) 2 ) + tan v i x ( αi 1 (x) Suppose G1 (max aspect ratio γ ) and G2 (min edge length d ) hold. Let λ i be the mean value coordinates. Then there exists C λ > 0 such that λ i H 1 (Ω) C λ 2 ) Proof: Divide analysis into six cases based on proximity to v a and size of α i and α j r a v a Case 1/2 x v i (Case 1) v j (Case 2) α i/α j v a Case 3/4 r a x v i (Case 3) v j (Case 4) α i/α j Case 5 v j r a v a x α j α i Case 6 v j v a r a x α j α i v i v i Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 14 / 45

Polygonal Finite Element Optimal Convergence Theorem 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 (aspect ratio) G2 (min edge length) G3 (max interior angle) Triangulated λ Tri - - S,N Wachspress λ Wach S S S,N Sibson λ Sibs S S - Mean Value λ MV S S - Harmonic λ Har 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, in press, 2013. Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 15 / 45

Outline 1 Linear Polygonal Elements with GBCs 2 Quadratic Serendipity Elements on Polygons 3 Cubic Hermite Serendipity Elements on Cubes 4 Current and Future Work Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 16 / 45

From linear to quadratic elements A naïve quadratic element is formed by products of linear GBCs: {λ i } pairwise products {λ aλ b } Why is this naïve? For an n-gon, this construction gives n + ( n 2) basis functions λaλ b 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 2n functions needed! Problem Statement Construct 2n basis functions associated to the vertices and edge midpoints of an arbitrary n-gon such that a quadratic convergence estimate is obtained. Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 17 / 45

Polygonal Quadratic Serendipity Elements We define matrices A and B to reduce the naïve quadratic basis. filled dot = Lagrangian domain point = all functions in the set evaluate to 0 except the associated function which evaluates to 1 open dot = non-lagrangian domain point = partition of unity satisfied, but not Lagrange property {λ i } n pairwise products {λ aλ b } n + ( ) n 2 A {ξ ij } 2n B {ψ ij } Linear Quadratic Serendipity Lagrange 2n Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 18 / 45

From quadratic to serendipity The bases are ordered as follows: ξ ii and λ aλ a = basis functions associated with vertices ξ i(i+1) and λ aλ a+1 = basis functions associated with edge midpoints λ aλ b = basis functions associated with interior diagonals, i.e. b / {a 1, a, a + 1} Serendipity basis functions ξ ij are a linear combination of pairwise products λ aλ b : ξ ii. ξ i(i+1) = A λ aλ a. λ aλ a+1. λ aλ b = c11 11 cab 11 c 11 (n 2)n............. c ij 11 c ij ab c ij (n 2)n............. c n(n+1) 11 c n(n+1) ab c n(n+1) (n 2)n λ aλ a. λ aλ a+1. λ aλ b Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 19 / 45

From quadratic to serendipity We require the serendipity basis to have quadratic approximation power: Constant precision: 1 = ξ ii + 2ξ i(i+1) i Linear precision: x = v i ξ ii + 2v i(i+1) ξ i(i+1) i Quadratic precision: xx T = v i v T i ξ ii + (v i v T i+1 + v i+1 v T i )ξ i(i+1) i ξ b(b+1) ξ bb ξ b(b 1) Theorem Constants {cij ab } exist for any convex polygon such that the resulting basis {ξ ij } satisfies constant, linear, and quadratic precision requirements. ξ a(a 1) ξ aa ξ a(a+1) λ a λ b Proof: We produce a coefficient matrix A with the structure A := [ I A ] where A has only six non-zero entries per column and show that the resulting functions satisfy the six precision equations. Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 20 / 45

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 λ 0 λ 0 λ 1 λ 1 1 ψ 00 ψ 11 ψ01 λ 0 λ 1 Pairwise products 1 Lagrange basis 1 Translation between these two bases is straightforward and generalizes to the higher dimensional case. Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 21 / 45

From serendipity to Lagrange {ξ ij } B {ψ ij } [ψ ij ] = ψ 11 ψ 22... ψ nn ψ 12 ψ 23... ψ n1 = 1 1 1 1 1 1.................. 1 1 1 4 4 0...... 4 ξ 11 ξ 22... ξ nn ξ 12 ξ 23... ξ n1 = B[ξ ij ]. Andrew Gillette - UCSD ( ) Polygonal & Serendipity FEM SMU - Feb 2013 22 / 45

Serendipity Theorem {λ i } pairwise products {λ aλ b } A {ξ ij } B {ψ ij } Theorem Given bounds on polygon aspect ratio (G1), minimum edge length (G2), and maximum interior angles (G3): A is uniformly bounded, B is uniformly bounded, and span{ψ ij } P 2 (R 2 ) = quadratic polynomials in x and y We obtain the quadratic a priori error estimate: u u h H 1 (Ω) C h 2 u H 3 (Ω) RAND, G., BAJAJ Quadratic Serendipity Finite Element on Polygons Using Generalized Barycentric Coordinates, Submitted, 2011 Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 23 / 45

Special case of a square v14 v4 v34 v3 v23 Bilinear functions are barycentric coordinates: λ 1 = (1 x)(1 y) λ 2 = x(1 y) λ 3 = xy λ 4 = (1 x)y v1 ξ 11 ξ 22 ξ 33 ξ 44 ξ 12 ξ 23 ξ 34 ξ 14 v12 = v2 Compute [ξ ij ] := [ ] I A [λaλ b ] λ 1 λ 1 λ 2 λ 2 λ 3 λ 3 λ 4 λ 4 λ 1 λ 2 = λ 2 λ 3 λ 3 λ 4 λ 1 λ 4 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1/2 1/2 0 0 1/2 1/2 0 0 1/2 1/2 0 1 1/2 1/2 (1 x)(1 y)(1 x y) x(1 y)(x y) xy( 1 + x + y) (1 x)y(y x) (1 x)x(1 y) x(1 y)y (1 x)xy (1 x)(1 y)y span { ξ ii, ξ i(i+1) } = span {1, x, y, x 2, y 2, xy, x 2 y, xy 2} =: S 2 (I 2 ) Hence, this provides a computational basis for the serendipity space S 2 (I 2 ) defined in ARNOLD, AWANOU The serendipity family of finite elements, Found. Comp. Math, 2011. Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 24 / 45

Numerical evidence for non-affine image of a square Instead of mapping, use quadratic serendipity GBC interpolation with mean value coordinates: n ( vi + v ) i+1 u h = I qu := u(v i )ψ ii + u ψ i(i+1) 2 n = 2 n = 4 i=1 Non-affine bilinear mapping u u h L 2 (u u h ) L 2 n error rate error rate 2 5.0e-2 6.2e-1 4 6.7e-3 2.9 1.8e-1 1.8 8 9.7e-4 2.8 5.9e-2 1.6 16 1.6e-4 2.6 2.3e-2 1.4 32 3.3e-5 2.3 1.0e-2 1.2 64 7.4e-6 2.1 4.96e-3 1.1 ARNOLD, BOFFI, FALK, Math. Comp., 2002 Quadratic serendipity GBC method u u h L 2 (u u h ) L 2 n error rate error rate 2 2.34e-3 2.22e-2 4 3.03e-4 2.95 6.10e-3 1.87 8 3.87e-5 2.97 1.59e-3 1.94 16 4.88e-6 2.99 4.04e-4 1.97 32 6.13e-7 3.00 1.02e-4 1.99 64 7.67e-8 3.00 2.56e-5 1.99 128 9.59e-9 3.00 6.40e-6 2.00 256 1.20e-9 3.00 1.64e-6 1.96 Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 25 / 45

Outline 1 Linear Polygonal Elements with GBCs 2 Quadratic Serendipity Elements on Polygons 3 Cubic Hermite Serendipity Elements on Cubes 4 Current and Future Work Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 26 / 45

Serendipity Spaces on Cubes in R d Tensor product methods on cubical meshes in R d are well known, but serendipity methods will converge at the same rate with fewer basis functions per element: tensor product elements O(h) O(h 2 ) O(h 3 ) O(h) O(h 2 ) O(h 3 ) serendipity elements Example: For O(h 3 ), d = 3, 50% fewer functions 50% smaller linear system Many modern applications use tensor product elements, often due to ease of implementation. Serendipity elements can reduce computation time, but implementation requires a geometric decomposition. Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 27 / 45

What is a geometric decomposition? A geometric decomposition for a finite element space is an explicit correspondence: {1, x, y, x 2, y 2, xy, x 3, y 3, x 2 y, xy 2, x 3 y, xy 3 } {ϑ 11, ϑ 14, ϑ 41, ϑ 44, ϑ 12, ϑ 13, ϑ 42, ϑ 43, ϑ 21, ϑ 31, ϑ 24, ϑ 34 } 14 24 34 44 13 43 12 42 11 21 31 41 monomials basis functions domain points Previously known basis functions employ Legendre polynomials These functions bear no symmetrical correspondence to the domain points 1 (x 1)(y 1) ( 1 3 4 4 2 x 2 1 ) (y 1) 1 5 x ( x 2 1 ) (y 1) 4 2 vertex edge (quadratic) edge (cubic) SZABÓ AND I. BABUŠKA Finite element analysis, Wiley Interscience, 1991. Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 28 / 45

Cubic Hermite Geometric Decomposition: 1D {1, x, x 2, x 3 } {ψ 1, ψ 2, ψ 3, ψ 4 } 1 2 3 4 approxim. geometry monomials basis functions domain points Cubic Hermite Basis on [0, 1] Approximation: x r = ψ 1 ψ 2 ψ 3 := ψ 4 1 3x 2 + 2x 3 x 2x 2 + x 3 x 2 x 3 3x 2 2x 3 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 4 ε r,i ψ i, for r = 0, 1, 2, 3, where [ε r,i ] = i=1 1 0 0 1 0 1 1 1 0 0 2 1 0 0 3 1 Geometry: u = u(0)ψ 1 + u (0)ψ 2 u (1)ψ 3 + u(1)ψ 4, u P 3 ([0, 1]) }{{} cubic polynomials Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 29 / 45

Cubic Hermite Geometric Decomposition: 2D { x r y s } } 0 r, s 3 {{ } Q 3 ([0, 1] 2 ) { ψi (x)ψ j (y) 1 i, j 4 } 14 24 34 44 13 23 33 43 12 22 32 42 11 21 31 41 monomials basis functions domain points Approximation: x r y s = 4 4 ε r,i ε s,j ψ ij, for 0 r, s 3, ε r,i as in 1D. i=1 j=1 Geometry: ψ 11 ψ 21 ψ 22 u = u (0,0) ψ 11 + xu (0,0) ψ 21 + yu (0,0) ψ 12 + x yu (0,0) ψ 22 +, u Q 3 ([0, 1] 2 ) Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 30 / 45

Cubic Hermite Geometric Decomposition: 3D { x r y s z t } } 0 r, s, t 3 {{ } Q 3 ([0, 1] 3 ) ψ i(x)ψ j (y)ψ k (z) 1 i, j, k 4 144 143 142 141 134 133 132 131 244 344 444 234 334 444 224 324 444 124 114 214 314 414 123 113 213 313 413 122 112 212 312 412 121 111 211 311 411 monomials basis functions domain points 4 4 4 Approximation: x r y s z t = ε r,i ε s,j ε t,k ψ ijk, for 0 r, s, t 3, ε r,i as in 1D. i=1 j=1 k=1 Geometry: Contours of level sets of the basis functions: ψ 111 ψ 112 ψ 212 ψ 222 Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 31 / 45

Two families of finite elements on cubical meshes Q r Λ k ([0, 1] n ) standard tensor product spaces early work: RAVIART, THOMAS 1976, NEDELEC 1980 more recently: ARNOLD, BOFFI, BONIZZONI arxiv:1212.6559, 2012 ( degree r in each variable) S r Λ k ([0, 1] n ) serendipity finite element spaces (superlinear degree r) early work: STRANG, FIX An analysis of the finite element method 1973 more recently: ARNOLD, AWANOU FoCM 11:3, 2011, and arxiv:1204.2595, 2012. The superlinear degree of a polynomial ignores linearly-appearing variables. Q 3 Λ 0 ([0,1] 2 ) (dim=16) {}}{ n = 2 : { 1, x, y, x 2, y 2, xy, x 3, y 3, x 2 y, xy 2, x 3 y, xy 3, x 2 y 2, x 3 y 2, x 2 y 3, x 3 y 3 } }{{} S 3 Λ 0 ([0,1] 2 ) (dim=12) Q 3 Λ 0 ([0,1] 3 ) (dim=64) {}}{ n = 3 : { 1,..., xyz, x 3 y, x 3 z, y 3 z,..., x 3 yz, xy 3 z, xyz 3, x 3 y 2,..., x 3 y 3 z 3 } }{{} S 3 Λ 0 ([0,1] 3 ) (dim=32) Q r Λ k and S r Λ k and have the same key mathematical properties needed for stability (degree, inclusion, trace, subcomplex, unisolvence, commuting projections) but for fixed k 0, r, n 2 the serendipity spaces have fewer degrees of freedom Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 32 / 45

Cubic Hermite Serendipity Geom. Decomp: 2D Theorem [G, 2012]: A Hermite-like geometric decomposition of S 3 ([0, 1] 2 ) exists. { x r y s } } sldeg 3 {{ } S 3 ([0, 1] 2 ) { ϑ lm (to be defined) } 14 24 34 44 13 43 12 42 11 21 31 41 monomials basis functions domain points Approximation: x r y s = lm ε r,i ε s,j ϑ lm, for superlinear degree(x r y s ) 3 Geometry: ϑ 11 ϑ 21 u = u (0,0) ϑ 11 + xu (0,0) ϑ 21 + yu (0,0) ϑ 12 + u S 3 ([0, 1] 2 ) Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 33 / 45

Cubic Hermite Serendipity Geom. Decomp: 2D Proof Overview: (akin to quadratic approach on polygons) 1 Fix index sets and basis orderings based on domain points: V = vertices (11, 14,...) E = edges (12, 13,...) D = interior (22, 23,...) 14 24 34 44 13 23 33 43 12 22 32 42 11 21 31 41 [ϑ lm ] := [ ϑ 11, ϑ 14, ϑ 41, ϑ 44, ϑ 12, ϑ 13, ϑ 42, ϑ 43, ϑ 21, ϑ 31, ϑ 24, ϑ 34 ], [ψ ij ] := [ ψ 11, ψ 14, ψ 41, ψ 44, ψ 12, ψ 13, ψ 42, ψ 43, ψ 21, ψ 31, ψ 24, ψ 34, ψ 22, ψ 23, ψ 32, ψ 33 ] }{{}}{{}}{{} indices in V indices in E indices in D 2 Define a 12 16 matrix H with entries h lm ij so that lm V E, ij V E D. 3 Define the serendipity basis functions ϑ lm via [ϑ lm ] := H[ψ ij ] and show that the approximation and geometry properties hold. Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 34 / 45

Cubic Hermite Serendipity Geom. Decomp: 2D Proof Details: 2 Define a 12 16 matrix H with entries hij lm so that lm V E, ij V E D. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 I 0 1 0 1 H := (12x12 identity matrix) 1 0 1 0 0 1 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 3 Define [ϑ lm ] := H[ψ ij ]. The geometry property holds since for lm V E, ϑ lm = ψ lm + }{{} ij D bicubic Hermite hij lm ψ ij }{{} zero on boundary = ϑ lm ψ lm on edges Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 35 / 45

Cubic Hermite Serendipity Geom. Decomp: 2D Proof Details: To prove that the approximation property holds, observe: [ϑ lm ] := H[ψ ij ] implies ij hij lm ψ ij = ϑ lm For all (r, s) pairs such that sldeg(x r y s ) 3, the matrix entries in column ij satisfy ε r,i ε s,j = ε r,l ε s,mhij lm lm V E Substitute these into the Hermite 2D approximation property: x r y s = ε r,l ε s,mhij lm ψ ij ε r,i ε s,j ψ ij = ij V E D ij = ε r,l ε s,m hij lm ψ ij = ε r,l ε s,mϑ lm lm lm ij Hence [ϑ lm ] is a basis for S 2 ([0, 1] 2 ), completing the geometric decomposition. lm Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 36 / 45

Hermite Style Serendipity Functions (2D) ϑ 11 (x 1)(y 1)( 2 + x + x 2 + y + y 2 ) ϑ 14 (x 1)(y + 1)( 2 + x + x 2 y + y 2 ) ϑ 41 (x + 1)(y 1)( 2 x + x 2 + y + y 2 ) ϑ 44 (x + 1)(y + 1)( 2 x + x 2 y + y 2 ) ϑ 12 (x 1)(y 1) 2 (y + 1) ϑ [ϑ lm ] = 13 = (x 1)(y 1)(y + 1) 2 ϑ 42 (x + 1)(y 1) 2 (y + 1) 1 8 ϑ 43 (x + 1)(y 1)(y + 1) 2 ϑ 21 ϑ (x 1) 2 (x + 1)(y 1) 31 ϑ (x 1)(x + 1) 2 (y 1) 24 (x 1) 2 (x + 1)(y + 1) ϑ 34 (x 1)(x + 1) 2 (y + 1) ϑ 11 ϑ 21 ϑ 31 Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 37 / 45

Cubic Hermite Serendipity Geom. Decomp: 3D Theorem [G, 2012]: A Hermite-like geometric decomposition of S 3 ([0, 1] 3 ) exists. { x r y s z t } } sldeg 3 {{ } S 3 ([0, 1] 3 ) { ϑ lmn (to be defined) } 244 344 444 144 444 134 444 143 124 114 214 314 414 142 113 413 141 131 112 412 121 111 211 311 411 monomials basis functions domain points Approximation: x r y s z t = lmn ε r,i ε s,j ε t,k ϑ lmn, for superlinear degree(x r y s z t ) 3 Geometry: u = u (0,0,0) ϑ 111 + xu (0,0,0) ϑ 211 + yu (0,0,0) ϑ 121 + zu (0,0,0) ϑ 112 + u S 3 ([0, 1] 3 ) ϑ 111 ϑ 112 Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 38 / 45

Cubic Hermite Serendipity Geom. Decomp: 3D Proof Overview: 1 Fix index sets and basis orderings based on domain points: V = vertices (111,...) E = edges (112,...) F = face interior (122,...) M = volume interior (222,...) 244 344 444 144 234 334 444 134 224 324 444 143 124 133 114 214 314 414 142 123 132 113 213 313 413 141 122 131 112 212 312 412 121 111 211 311 411 [ϑ lmn ] := [ ϑ 111,..., ϑ 444, ϑ 112,..., ϑ 443 ], [ψ ijk ] := [ ψ 111,..., ψ 444, ψ 112,..., ψ 443, ψ 122,..., ψ 433, ψ 222,..., ψ 333 ] }{{}}{{}}{{}}{{} indices in V indices in E indices in F indices in M 2 Define a 32 64 matrix W with entries h lmn ijk (where lmn V E) 3 Define the serendipity basis functions ϑ lmn via [ϑ lmn ] := W[ψ ijk ] and show that the approximation and geometry properties hold. Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 39 / 45

Cubic Hermite Serendipity Geom. Decomp: 3D Proof Details: 2 Define a 32 64 matrix W with entries hijk lmn W := I (32x32 identity matrix) 3 Define [ϑ lmn ] := W[ψ ijk ]. so that lmn V E specific full rank 32x32 matrix with entries -1, 0, or 1 Confirm directly that [ϑ lmn ] restricts to [ϑ lm ] on faces. Similar proof technique confirms geometry and approximation properties. ϑ 111 (x 1)(y 1)(z 1)( 2 + x + x 2 + y + y 2 + z + z 2 ) ϑ 114 (x 1)(y 1)(z + 1)( 2 + x + x 2 + y + y 2 z + z 2 ) [ϑ lmn ] =. = 1. 16 ϑ 442 (x + 1)(y + 1)(z 1) 2 (z + 1) ϑ 443 (x + 1)(y + 1)(z 1)(z + 1) 2 Complete list and more details in my paper: GILLETTE Hermite and Bernstein Style Basis Functions for Cubic Serendipity Spaces on Squares and Cubes, arxiv:1208.5973, 2012 Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 40 / 45

Cubic Bernstein Serendipity Geom. Decomp: 2D, 3D Cubic Bernstein Basis on [0, 1] ψ 1 (1 x) 3 ψ 2 ψ 3 := (1 x) 2 x (1 x)x 2 ψ 4 x 3 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 Theorem [G, 2012]: Bernstein-like geometric decompositions of S 3 ([0, 1] 2 ) and S 3 ([0, 1] 3 ) exist. ξ 11 (x 1)(y 1)( 2 2x + x 2 2y + y 2 ) ξ 14 (x 1)(y + 1)( 2 2x + x 2 + 2y + y 2 ) [ξ lm ] = 1. =. 16 ξ 24 (x 1) 2 (x + 1)(y + 1) ξ 34 (x 1)(x + 1) 2 (y + 1) ξ 111 (x 1)(y 1)(z 1)( 5 2x + x 2 2y + y 2 2z + z 2 ) ξ 114 (x 1)(y 1)(z + 1)( 5 2x + x 2 2y + y 2 + 2z + z 2 ) [ξ lmn ] = 1. =. 32 ξ 442 (x + 1)(y + 1)(z 1) 2 (z + 1) ξ 443 (x + 1)(y + 1)(z 1)(z + 1) 2 Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 41 / 45

Bernstein Style Serendipity Functions (2D) β 11 β 21 β 31 ξ 11 ξ 21 ξ 31 1.0 0.8 0.6 0.4 0.2 Bicubic Bernstein functions (top) and Bernstein-style serendipity functions (bottom). Note boundary agreement with Bernstein functions. 0.0 0.2 0.4 0.6 0.8 1.0 Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 42 / 45

Outline 1 Linear Polygonal Elements with GBCs 2 Quadratic Serendipity Elements on Polygons 3 Cubic Hermite Serendipity Elements on Cubes 4 Current and Future Work Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 43 / 45

Application: Cardiac Electrophysiology Cubic Hermite serendipity functions recently incorporated into Continuity software package for cardiac electrophysiology models. Used to solve the monodomain equations, a type of reaction-diffusion equations Initial results show agreement of serendipity and standard bicubics on a benchmark problem with a 4x computational speedup in 3D. Fast computation essential to clinical applications and real time simulations GONZALEZ, VINCENT, G., MCCULLOCH High Order Interpolation Methods in Cardiac Electrophysiology Simulation, in preparation, 2013. Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 44 / 45

Acknowledgments Michael Holst National Biomedical Computation Resource Matt Gonzalez Kevin Vincent Andrew McCulloch Chandrajit Bajaj Alexander Rand UC San Diego UC San Diego UC San Diego UC San Diego UC San Diego UT Austin UT Austin / CD-adapco Thanks for the opportunity to speak! Slides and pre-prints: http://ccom.ucsd.edu/~agillette Andrew Gillette - UCSD Polygonal ( ) & Serendipity FEM SMU - Feb 2013 45 / 45