Geometric Decomposition of Serendipity Finite Element Spaces

Transcription

1 Geometric Decomposition of Serendipity Finite Element Spaces Andrew Gillette Department of Mathematics University of California, San Diego agillette/ Joint Mathematics Meetings 2013 AMS Special Session Finite Element Exterior Calculus and Applications Andrew Gillette - UCSD Geometric ( ) Decomposition of SFE JMM - Jan / 25

2 What is a serendipity finite element method? Goal: Efficient, accurate approximation of the solution to a PDE over Ω R n. Standard O(h r ) tensor product finite element method in R n : Mesh Ω by n-dimensional cubes of side length h. Set up a linear system involving (r + 1) n degrees of freedom (DoFs) per cube. For unknown continuous solution u and computed discrete approximation u h : u u h H 1 (Ω) C h r u H }{{} r+1 (Ω), u H r+1 (Ω). }{{} approximation error optimal error bound A O(h r ) serendipity FEM converges at the same rate with fewer DoFs per element: O(h) O(h 2 ) O(h 3 ) O(h) O(h 2 ) O(h 3 ) tensor product elements serendipity elements Example: For O(h 3 ), d = 3, 50% fewer DoFs 50% smaller linear system Andrew Gillette - UCSD Geometric ( ) Decomposition of SFE JMM - Jan / 25

3 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 } 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) ( 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, Andrew Gillette - UCSD Geometric ( ) Decomposition of SFE JMM - Jan / 25

4 Motivations and Related Topics Goal: Construct geometric decompositions of serendipity spaces using linear combinations of standard tensor product functions. Focus: Cubic Hermites. Modern mathematics: Finite Element Exterior Calculus, Discrete Exterior Calculus, Virtual Element Methods... ARNOLD, AWANOU The serendipity family of finite elements, Found. Comp. Math, Isogeometric analysis: Finding basis functions suitable for both domain description and PDE approximation avoids the expensive computational bottleneck of re-meshing. COTTRELL, HUGHES, BAZILEVS Isogeometric Analysis: Toward Integration of CAD and FEA, Wiley, Computational Efficiency: Modern applications such as patient-specific cardiac electrophysiology need efficient, stable, error-bounded real-time methods Flexible Domain Meshing: Serendipity type elements for Voronoi meshes provides various computational benefits. RAND, GILLETTE, BAJAJ Quadratic Serendipity Finite Elements on Polygons Using Generalized Barycentric Coordinates arxiv: , Andrew Gillette - UCSD Geometric ( ) Decomposition of SFE JMM - Jan / 25

5 Table of Contents 1 The Cubic Case: Hermite Functions, Serendipity Spaces 2 Geometric Decompsitions of Cubic Serendipity Spaces 3 Applications and Future Directions Andrew Gillette - UCSD Geometric ( ) Decomposition of SFE JMM - Jan / 25

6 Outline 1 The Cubic Case: Hermite Functions, Serendipity Spaces 2 Geometric Decompsitions of Cubic Serendipity Spaces 3 Applications and Future Directions Andrew Gillette - UCSD Geometric ( ) Decomposition of SFE JMM - Jan / 25

7 Cubic Hermite Geometric Decomposition: 1D {1, x, x 2, x 3 } {ψ 1, ψ 2, ψ 3, ψ 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 ε r,i ψ i, for r = 0, 1, 2, 3, where [ε r,i ] = i= 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 Geometric ( ) Decomposition of SFE JMM - Jan / 25

8 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 } 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 Geometric ( ) Decomposition of SFE JMM - Jan / 25

9 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 monomials basis functions domain points 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 Geometric ( ) Decomposition of SFE JMM - Jan / 25

10 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: , 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: , 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 FEEC (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 Geometric ( ) Decomposition of SFE JMM - Jan / 25

11 Outline 1 The Cubic Case: Hermite Functions, Serendipity Spaces 2 Geometric Decompsitions of Cubic Serendipity Spaces 3 Applications and Future Directions Andrew Gillette - UCSD Geometric ( ) Decomposition of SFE JMM - Jan / 25

12 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) } 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 Geometric ( ) Decomposition of SFE JMM - Jan / 25

13 Cubic Hermite Serendipity Geom. Decomp: 2D Proof Overview: 1 Fix index sets and basis orderings based on domain points: V = vertices (11, 14,...) E = edges (12, 13,...) D = interior (22, 23,...) [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 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 Geometric ( ) Decomposition of SFE JMM - Jan / 25

14 Cubic Hermite Serendipity Geom. Decomp: 2D Proof Details: 2 Define a matrix H with entries hij lm so that lm V E, ij V E D I H := (12x12 identity matrix) 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 Geometric ( ) Decomposition of SFE JMM - Jan / 25

15 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 Geometric ( ) Decomposition of SFE JMM - Jan / 25

16 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 Geometric ( ) Decomposition of SFE JMM - Jan / 25

17 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) } 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 xu (0,0,0) ϑ yu (0,0,0) ϑ zu (0,0,0) ϑ u S 3 ([0, 1] 3 ) ϑ 111 ϑ 112 Andrew Gillette - UCSD Geometric ( ) Decomposition of SFE JMM - Jan / 25

18 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,...) [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 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 Geometric ( ) Decomposition of SFE JMM - Jan / 25

19 Cubic Hermite Serendipity Geom. Decomp: 3D Proof Details: 2 Define a 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 ] =. = 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: , 2012 Andrew Gillette - UCSD Geometric ( ) Decomposition of SFE JMM - Jan / 25

20 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 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 Geometric ( ) Decomposition of SFE JMM - Jan / 25

21 Bernstein Style Serendipity Functions (2D) β 11 β 21 β 31 ξ 11 ξ 21 ξ Bicubic Bernstein functions (top) and Bernstein-style serendipity functions (bottom). Note boundary agreement with Bernstein functions Andrew Gillette - UCSD Geometric ( ) Decomposition of SFE JMM - Jan / 25

22 Outline 1 The Cubic Case: Hermite Functions, Serendipity Spaces 2 Geometric Decompsitions of Cubic Serendipity Spaces 3 Applications and Future Directions Andrew Gillette - UCSD Geometric ( ) Decomposition of SFE JMM - Jan / 25

23 Application: Cardiac Electrophysiology Initial implementation of cubic Hermite serendipity functions as part of the Continuity software package for cardiac electrophysiology models. Used to solve the monodomain equations, similar to the heat equation: tu u = f Cubic Hermite serendipity functions resolve the voltage wavefront equivalently (in eyeball norm) to regular Hermite bicubics. Initial results show a 4x computational speedup in 3D. Current research into isogeometric analysis with applications to clinical modeling. Andrew Gillette - UCSD Geometric ( ) Decomposition of SFE JMM - Jan / 25

24 Future Directions and Open Questions Construction of Bernstein-like bases for S r Λ k ([0, 1] n ) for Higher order scalar cases (k = 0, r > 3, n = 2, 3) Higher form order cases (k > 0) Generalized Bernstein-like construction process: [ϑ l ] := M [β i ] [ basis for S r Λ k ([0, 1] n ) ] := M [ Bernstein basis for Q r Λ k ([0, 1] n ) ] Define entries of M so that the appropriate approximation and geometry properties hold for [ϑ s]. Andrew Gillette - UCSD Geometric ( ) Decomposition of SFE JMM - Jan / 25

25 Acknowledgments Matt Gonzalez Andrew McCulloch Michael Holst National Biomedical Computation Resource UC San Diego UC San Diego UC San Diego UC San Diego Slides and pre-prints: Andrew Gillette - UCSD Geometric ( ) Decomposition of SFE JMM - Jan / 25