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Generalized Barycentric Coordinate Finite Element Methods on Polytope Meshes

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Generalized Barycentric Coordinate Finite Element Methods on Polytope Meshes 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 2 H2(D) Z Z weak form ru · rφ = f φ 8φ 2 H1(D) D D Z Z 1 discrete form ruh · rφh = f φh 8φh 2 Vh finite dim. ⊂ H (D) D D Typical finite element method: ! Mesh D by polytopes fPg with vertices fvi g; 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 jju − uhjjH1(P) ≤ Ch jujHp+1(P); 8u 2 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; 8 L : P ! R linear. v2V Familiar properties are implied by this definition: X X λv ≡ 1 vλv(x) = x λvi (vj ) = δij v2V v2V | {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. fλi g fλi λj g f ij g 3 Construction of H(curl) and H(div) methods using λv and rλv functions. grad curl div H1 / H(curl) / H(div) / L2 fλi g fλi rλj g fλi rλj × rλk g fχP g 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) juj 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 jrλv(x)j x2P v2V 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 jjujjC0(P) ≤ CS jjujjHk (P) independent of u 2 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 jrλv(x)j x2P v2V 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: jrλvj large () interior angle at v is large () the altitude “at v” is small For Wachspress coordinates, we generalize to polygons: jrλvj 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 jrλv(x)j =:Λ is large () h∗ is small x2P v2V 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 v62f 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 fp (x)g = 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 jrλv(x)j =:Λ ≤ where h∗ := min min hf (v). x h∗ f v62f v 1 Bound jrλvj by summations over faces incident and not incident to v. ! ! X 1 X X 1 X jrλvj ≤ λv 1 − λu + λv λu hf hf f 2Fv u2f f 62Fv u2f 2 Summing over v gives a constant bound. ! ! X X 1 X X jrλvj ≤ 2 1 − λu λu hf v f 2F u2f u2f 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 jff : f 3 vgjλv = h∗ h∗ h∗ f 2F u2f v2V 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 62 f . 2 Let v be the vertex in f closest to w. Show that v f 1 jrλw(v)j = hf (w) w 3 Conclude the result, since 1 1 Λ ≥ jrλw(v)j = = 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 jrλv(x)j. x2P v d 1 2d simple convex polytope in R ≤ Λ ≤ h∗ h∗ d 1 d + 1 d-simplex in R ≤ Λ ≤ h∗ h∗ p 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!1 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 (2; 1) ρ(Ω) Three possible geometric conditions on a polygonal mesh: G1. BOUNDED ASPECT RATIO: 9 γ∗ < 1 such that γ < γ∗ G2.
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