Outline. 1 Motivating examples. 2 Hilbert complexes and their discretization. 3 Finite element differential forms. 4 The elasticity complex 1 / 25

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1 Outline 1 Motivating examples Hoge theory, Hilbert complexes, an finite element ifferential forms Douglas N. Arnol 2 Hilbert complexes an their iscretization 3 Finite element ifferential forms 4 The elasticity complex ICIAM minisymposium on Applie Hoge Theory, 19 July 2011 Joint with R. Falk an R. Winther. Primary reference: Finite element exterior calculus: From Hoge theory to numerical stability, Bull. AMS 2010, pp FEEC: Hilbert complex framework + finite element iff l forms + applications 1 / 25 2 / 25 Vector Poisson equation on a plane omain u gra u = f in (mo H), u n = 0, u n = 0, u H on Motivating example Just the Hoge Laplacian for 1-forms: u + u = f (mo H) Weak formulation: fin u H() H() H such that ( u v+ u v) x = f v x, v H() H() H Variational formulation: u = arg min H() H() H ( 1 2 u 2 + u 2 x f u x) 3 / 25

2 Stanar finite elements o not work f = (0, x) P1 elements Hilbert complexes an their iscretization f = ( 1, 0) P1 elements 4 / 25 Hilbert complexes Properties of close Hilbert complexes We view the exterior erivative as a close unboune operator L 2 Λ k L 2 Λ k+1 with omain Ajoint complex: k is ensely-efine, close, w/ close range HΛ k = { u L 2 Λ k u L 2 Λ k+1 }. Resulting structure is a close Hilbert complex, which abstracts the e Rham complex: Hilbert spaces W 0, W 1,..., W n ; Densely efine close operators W k k W k+1 with omain V k W k an close range B k+1, satisfying: k 1 k = 0 (i.e., B k Z k := ker k V k ) Defining v 2 = V k v 2 + W k v 2 W k+1, we get a complex of Hilbert spaces 0 V 0 V 1 V n 0 0 V 0 Abstract Hoge Laplacian: Harmonic forms: Hoge ecomposition: V 1 V n 0 + : W k W k Z k /B k Z k B k = ker( + ) := H k W k = B k H k B k }{{}}{{} Z k Z k Poincaré inequality: c such that u c u u Z k V k with associate cohomology spaces Z k /B k 5 / 25 6 / 25

3 Mixe formulation of the (abstract) Hoge Laplacian σ = u, σ + u = f (mo H), u H Weak formulation: Given f W k, fin σ V k 1, u V k, p H k : σ, τ τ, u = 0 τ V k 1 σ, v + u, v + p, v = f, v v V k u, q = 0 q H k Variational formulation: 1 2 σ, σ 1 u, u σ, u u, p + f, u sale point 2 Well-poseness of the mixe formulation Theorem: (σ, u, p) V k 1 V k H k (τ, v, q) V k 1 V k H k 1. Boune: τ V + v V + q C( σ V + u V + p ) 2. Coercing: B c( σ 2 V + u 2 V + p 2 ) where B := σ, τ τ, u + σ, v + u, v + p, v u, q. Hoge ecompose u = η + s + z with η (Z k 1 ), s H k, z (Z k ) Choose τ = σ ɛη, v = u + σ + p, q = p s: B = σ 2 + σ 2 + ɛ η 2 + s 2 + u 2 + p 2 ɛ σ, η. ɛ σ, η 1 2 σ 2 + ɛ2 2 η 2 ; an, by Poincaré ineq, η cp η, so ɛ = c 2 P = B(σ, u, p; τ, v, q) c( σ 2 + σ 2 + η 2 + s 2 + u 2 V + p 2 ). But, Poincaré ineq also gives z cp z = cp u. 7 / 25 8 / 25 Discretization We now want to iscretize the mixe formulation with f.. subspaces V k h V k inexe by h (Galerkin). Of course we assume inf v vh V 0 as h 0 v V (A) vh V k h It turns out that there are two more key assumptions. Subcomplex assumption (SC): (Vh k) V k+1 h The subcomplex k 1 Vh k k V k+1 k+1 h structurepreserving iscretization is itself a Hilbert complex so we have iscrete harmonic forms H k h, iscrete Hoge ecomp, an iscrete Poincaré ineq with constant cp,h. Boune Cochain Projection assumption (BCP): V k k V k+1 πh k π k+1 h V k h k V k+1 h π k h : V k V k h πh k V -boune, uniform in h π k h a projection π k+1 h k = k πh k 9 / 25 Stability theorem Theorem Let (V k, k ) be a Hilbert complex an Vh k finite imensional subspaces satisfying A, SC, an BCP. Then πh inucesanisomorphismoncohomologyforh small gap ( H k, H k h) 0 The iscrete Poincaré inequality ω c ω, ω Z k h, hols with c inepenent of h Galerkin s metho is stable Proof of iscrete Poincaré inequality: Given ω Z k h, efine η Z k V k by η = ω. By the Poincaré inequality, η cp ω, whence η V c ω, so it is enough to show that ω c η V. Now, ω πhη Vh k an, by SC an BCP, (ω πhη) = ω πhω = 0, so ω πhη Z k h. Thus ω (ω πhη), so ω πhη by Pythagoras. Result follows since πh is boune. Note cp,h+ (cp 2 + 1)1/2 πh. 10 / 25

4 Convergence of Galerkin s metho Since we have uniform control of the Poincaré constant cp,h, the well-poseness argument applie on the iscrete level gives stability. From stability we get the basic energy estimate, which is quasi-optimal plus a small consistency error term if there are harmonic forms. Notation: for w V k, Theorem Assume SC an BCP. E(w) := inf w V k h w v V Then σ σh V + u uh V + p ph V c[ E(σ)+E(u)+E(p)+ɛ ] where ɛ inf E(PBu) sup E(r). v Vh k r H k r =1 Finite element ifferential forms Improve estimates can then be erive using uality techniques. 11 / 25 FE ifferential forms, the concrete realization of FEEC How to construct FE subspaces of HΛ k satisfying SP an BCP? FEEC reveals that there are precisely two natural families: triangulation form egree PrΛ k (T ) an P r Λ k (T ) polynomial egree Like all finite element spaces they are constructe from shape functions an egrees of freeom on each simplex T. Shape fns for PrΛ k : polynomial k-forms of egree r. Shape fns for P r Λ k efine via Koszul ifferential κ : Λ k+1 Λ k : (κω)x(v1,..., vk) = ωx(x, v1,, vk) P r Λ k (T ) = Pr 1Λ k (T ) + κ Pr 1Λ k+1 (T ) Degrees of freeom DOF for PrΛ k (T ): to a subsimplex f of imension we associate ω trf ω η, η P r+k Λ k (f) f Theorem. These DOFs are unisolvent an the resulting finite element space satisfies PrΛ k (T ) = { ω HΛ k () : ω T PrΛ k (T ) T T } DOF for P r Λ k (T ) (Hiptmair 99): ω trf ω η, η Pr+k 1Λ k (f) f + similar theorem / / 25

5 The Pr Λk family in 2D The Pr Λk family in 3D Pr Λ0 Pr Λ1 Pr Λ2 Pr Λ0 Pr Λ1 Pr Λ2 Pr Λ3 Lagrange Raviart-Thomas DG Lagrange Ne e lec ege I R-T-N face DG r =1 r =1 r =2 r =2 r =3 r =3 14 / 25 The Pr Λk family in 2D 15 / 25 Finite element e Rham subcomplexes Pr Λ0 Pr Λ1 Pr Λ2 Lagrange BDM DG We on t only want spaces, we also want them to fit together into iscrete e Rham complexes with BCP. One such FER subcomplex uses Pr Λk spaces of constant egree r : 0 Pr Λ0 (T ) Pr Λ1 (T ) Pr Λn (T ) 0 r =1 0 gra 0 Whitney 57 r =2 0 gra 0 Another uses Pr Λk spaces with ecreasing egree: r =3 0 Pr Λ0 (T ) Pr 1 Λ1 (T ) Pr n Λn (T ) 0 Sullivan 75 These are extreme cases. For every r 16 / 25 2n 1 such FER subcomplexes. 17 / 25

6 Finite element e Rham subcomplexes on cubes Tensor-prouct Serenipity (DNA-Awanou 11) The elasticity complex 18 / 25 Mixe finite elements for elasticity Fin stress σ : S = Rsym n n, isplacement u : V = Rn such that Aσ = ɛ(u), σ = f ) ( 1 A σ :σ + σ u + f u 2 σ,u x sale point H(;S) L 2 (V) Search for stable finite elements ates back to the 60s, very limite success. [Mixe nite elements were] achieve by Raviart an Thomas in the case of the heat conuction problem iscusse previously. Extension to the full stress problem is icult an as yet such elements have not been successfully note. Zienkiewicz, Taylor, Zhu The Finite Element Metho: Its Basis & Funamentals, 6th e., 2005 With FEEC, this is no longer true. Now, lots of elements. The elasticity complex in 2D The elasticity system is the n-hoge Laplacian associate to a complex: Airy potential stress loa 0 H 2 J () H(, ; S) L 2 (; R 2 ) 0 ( ) 2 Jφ = y φ xyφ, the Airy stress function, is secon orer! xyφ x 2 φ The question is: how to iscretize this sequence? This simplest element (DNA-Winther 02) involves 21 stress egrees of freeom. It provies a Hilbert subcomplex with boune cochain projections. 0 J 0 19 / / 25

7 The elasticity complex in 3D isplacement strain stress loa ɛ 0 H 1 (; R 3 J ) H(J, ; S) H(, ; S) L 2 (; R 3 ) 0 J = T The simplest FE subcomplex involves 162 DOFs for stress (DNA-Awanou-Winther 08). Much simpler nonconforming elements can be evise: 12 stress DOF in 2D (DNA-Winter 03), 36 in 3D (DNA-Awanou-Winther 11) 0 J 0 21 / 25 The elasticity complex with weak symmetry To obtain simpler conforming elements we went back to an ol iea, enforcing the symmetry of the stress tensor, skw σ = 0, via a Lagrange multiplier (Fraeijs e Veubeke 65, Amara-Thomas 79, DNA-Brezzi-Douglas 84). ( ) 1 σ,u A σ :σ + σ u + f u x 2 S.P. H(;S) L 2 (V) ( ) 1 σ,u,p A σ :σ + σ u + σ :p + f u x 2 S.P. H(;M) L 2 (V) L 2 (K) The associate elasticity complex: isplacement rotation strain 0 H 1 (; R 3 ) L 2 (gra, I) J (, K) H(J, ; M) ( ) skw J H(, ; M) L 2 (; R 3 ) L 2 (; K) 0 stress loa couple 22 / 25 The key to iscretizations of the elasticity complex The simplest choice The elasticity complex can be erive from the e Rham complex through a form of the BGG resolution. BGG can be applie to finite element e Rham subcomplexes, to get finite element subcomplexes of the elasticity complex. 0 gra 0 Theorem (AFW 06, 07): Choose two subcomplexes of the e Rham complex satisfying BCP: 0 V 0 h 0 Ṽ 0 h gra V 1 h gra Ṽ 1 h V 2 h Ṽ 2 h V 3 h 0 Ṽ 3 h 0 0 gra 0 Suppose that satisfy a surjectivity hypothesis. (Roughly, for each DOF of Vh 2 there is a corresponing DOF of Ṽ h 1.) stress: Ṽh 2(R3 ) Then isplacement: Ṽh 3 (R3 ) is a stable element choice. rotation: Vh 3(K) σ u p 23 / / 25

8 Features of the new mixe elements Base on HR formulation with weak symmetry; very natural Lowest egree element is very simple: full P1 for stress, P0 for isplacement an rotation Works for every polynomial egree Works the same in 2 an 3 (or more) imensions Robust to material constraints like incompressibility Provably stable an convergent Has been wiely implemente. Elastostatics, elastoynamics, viscoelasticity,... Have opene the way for many other elements: Awanou, Boffi, Brezzi, Cockburn, Demkowicz, Fortin, Gopalakrishnan, Guzman, Qiu, / 25