Application of the Finite Element Exterior Calculus to the Equations - - PowerPoint PPT Presentation

Application of the Finite Element Exterior Calculus to the Equations of Linear Elasticity

Richard S. Falk

Department of Mathematics Rutgers University

June 12, 2012 Joint work with: Douglas Arnold, University of Minnesota Ragnar Winther, Centre of Mathematics for Applications, University of Oslo, Norway

Richard S. Falk Finite Element Methods for Linear Elasticity

Outline Of Talk

◮ Variational formulations of the equations of linear elasticity ◮ Stability of discretizations of saddle-point problems ◮ Connections to exact sequences – continuous and discrete ◮ Exact sequences for elasticity ◮ From de Rham to elasticity ◮ Stability of continuous formulation of elasticity with weakly

imposed symmetry

◮ Finite element methods for the equations of elasticity from

connections to de Rham

Richard S. Falk Finite Element Methods for Linear Elasticity

Equations of linear elasticity

For N = 2, 3, equations of linear elasticity written as system: Aσ = εu, div σ = f in Ω ⊂ RN.

• stressfield σ(x) ∈ S (symmetric matrices).
• displacement field u(x) ∈ RN.
• f = f (x) given body force.
• A = A(x) : S → S given, uniformly positive definite, compliance

tensor (material dependent).

• ǫij(u) = [∂ui/∂xj + ∂uj/∂xi]/2.
• div of matrix field taken row-wise.
• If body clamped on boundary ∂Ω of Ω, BC: u = 0 on ∂Ω.

Richard S. Falk Finite Element Methods for Linear Elasticity

Stress–displacement formulations

Strongly imposed symmetry: Find (σ, u) ∈ H(div, Ω, S) × L2(Ω, RN) such that: (Aσ, τ) + (div τ, u) = 0, τ ∈ H(div, Ω, S), (div σ, v) = (f , v) v ∈ L2(Ω, RN). Weakly imposed symmetry: Find (σ, u, p) ∈ H(div, Ω, M) × L2(Ω, RN) × L2(Ω, K) such that: (Aσ, τ) + (div τ, u) + (τ, p) = 0, τ ∈ H(div, Ω, M), (div σ, v) = (f , v), v ∈ L2(Ω, RN), (σ, q) = 0, q ∈ L2(Ω, K). M = N × N matrices, K = skew symmetric matrices.

Richard S. Falk Finite Element Methods for Linear Elasticity

A simpler problem

Consider mixed formulation of Poisson’s equation ∆p = f . (u, v) + (p, div v) = 0 ∀v ∈ H(div, Ω, RN), (div u, q) = (f , q) ∀q ∈ L2(Ω, R). Approximation well understood and related to commuting diagrams for de Rham sequence. In 2-D, if we have finite element spaces and bounded projection operators satisfying commuting diagram: 0 − − → H1

curl

− − → H(div)

div

− − → L2 − − → 0   Π1

h

  Πd

h

  Π0

h

0 − − → Sh

curl

− − → Vh

div

− − → Qh − − → 0 then mixed finite element stable, and get quasi-optimal approximation.

Richard S. Falk Finite Element Methods for Linear Elasticity

Elasticity (Hilbert) complexes with strong symmetry

Find (σ, u) ∈ H(div, Ω, S) × L2(Ω, RN) such that (Aσ, τ) + (u, div τ) = 0 ∀τ ∈ H(div, Ω, S), (div σ, v) = (f , v) ∀v ∈ L2(Ω, RN). Corresponding complexes in this case: 0 − → H1(R3) ε − → H(J, S) J − → H(div, S) div − − → L2(R3) → 0, in 3-D, where Jσ = curl(curl σ)T. 0 − → H2 J − → H(div, S) div − − → L2(R2) → 0 in 2-D, where Jq = ∂2q/∂y2 −∂2q/∂x∂y −∂2q/∂x∂y ∂2q/∂x2

• .

Richard S. Falk Finite Element Methods for Linear Elasticity

Approximation in 2D

Although elasticity problem only involves last two spaces in complex 0 − → H2 J − → H(div, S) div − − → L2(R2) → 0, having full sequence gives clue how to choose discretization. One looks for subcomplex of form 0 − → Qh

J

− → Σh

div

− − → Vh → 0. For example, looking for simple finite element subspace of H2, one is led to choosing Qh = Argyris space of C 1 quintics. Since JQh ⊂ Σh, Σh must be a piecewise cubic space, and since Argyris space has 2nd derivative DOF at vertices, DOF for Σh will include vertex DOF (not usual for H(div) spaces).

Richard S. Falk Finite Element Methods for Linear Elasticity

Arnold-Winther elements

In 2002, Arnold-Winther constructed commuting diagrams of form: 0 − − → C 2(R)

J

− − → C 0(S)

div

− − → L2(R2) − − → 0   I 2

h

  I d

h

  I 0

h

0 − − → Qh

J

− − → Σh

div

− − → Vh − − → 0 Simplest case of family of elements: Qh = Argyris space of C 1

• quintics. Stress space Σh = p. cubic functions with p. linear

divergence (24 DOF). Displacements Vh = p. linear functions. However, since I 2

h involves point values of 2nd derivative and I 1 h

involves point values, these operators do not extend to bounded

• perators in Hilbert spaces H2 and H(div, S).

In Bulletin, bounded cochain projections constructed. Gives stability for corresponding mixed finite element method by Hilbert complex theory (assumes Ω star-shaped).

Richard S. Falk Finite Element Methods for Linear Elasticity

Elasticity with weakly imposed symmetry

In 2-D, setting W = K × R2, relevant complex: · · · − → C ∞(K) J − → C ∞(M) (skw

div)

− − − → C ∞(W) → 0. In 3-D, with W = K × R3, relevant complex: · · · − → C ∞(M) J − → C ∞(M) (skw

div)

− − − → C ∞(W) → 0. Here J : C ∞(M) → C ∞(M) denotes extension of previous

• perator.

Jτ = curl S−1 curl τ, S algebraic

Richard S. Falk Finite Element Methods for Linear Elasticity

New approach to discretization of elasticity sequences:

◮ Use procedure on continuous level to derive elasticity

sequence from multiple copies of de Rham sequence.

◮ Use this connection to establish stability for continuous

formulation of elasticity

◮ To discretize, start from known good discretizations of de

Rham sequence.

◮ Determine conditions so that an analogue of stability proof for

continuous problem will give stability of discrete problem. To see structure more clearly, adopt notation of differential forms. For simplicity, mostly consider 2D examples.

Richard S. Falk Finite Element Methods for Linear Elasticity

de Rham sequences with values in a vector space

Write 2-D de Rham sequence in form: 0 − → Λ0 d0 − → Λ1 d1 − → Λ2 → 0. Also consider sequences whose values lie in either V = Rn or K, space of skew-symmetric matrices. Both corresponding de Rham sequences also exact, e.g., 0 − → Λ0(V) d0 − → Λ1(V) d1 − → Λ2(V) → 0. Here Λk(V) consists of elements of form: ω(x) =

• I

fI(x)dxI with coefficients fI ∈ C ∞(Ω, V).

Richard S. Falk Finite Element Methods for Linear Elasticity

Weak symmetry elasticity sequence from de Rham (BGG)

Following ideas of Eastwood: Start from two de Rham sequences: · · · − → Λn−2(K) dn−2 − − − → Λn−1(K) dn−1 − − − → Λn(K) → 0, · · · − → Λn−2(V) dn−2 − − − → Λn−1(V) dn−1 − − − → Λn(V) → 0. For both n = 2 and n = 3, spaces Λn−1(V) are spaces of stresses and can be identified with n × n matrices. Let X = (x1, . . . , xn)T and define Kk : Λk(V) → Λk(K) by Kkω = XωT − ωX T Then define Sk := dkKk − Kk+1dk : Λk(V) → Λk+1(K)

Richard S. Falk Finite Element Methods for Linear Elasticity

The operator Sk

Can show: Sk is an algebraic operator. Two important operators: Sn−2 and Sn−1. Operator Sn−1 can be identified with skw, i.e., taking skew part of matrix (i.e., (W − W T)/2). n = 2 : Sn−2 ω1 ω2

• =

ω2 −ω2

• dx1 +

−ω1 ω1

• dx2

Easy to check S0 invertible. For n = 3, S1 more complicated, but still algebraic and invertible. Key property used to establish stability: dn−1Sn−2 = −Sn−1dn−2 n = 2 : (div W ) −1 1

• + 2 skw curl W = 0.

Much more complicated identity in 3-d.

Richard S. Falk Finite Element Methods for Linear Elasticity

Elasticity sequence from de Rham sequence

Picture is: · · · − → Λn−2(K) dn−2 − − − → Λn−1(K) dn−1 − − − → Λn(K) → 0 ր Sn−2 ր Sn−1 · · · − → Λn−2(V) dn−2 − − − → Λn−1(V) dn−1 − − − → Λn(V) → 0. Since Sn−2 invertible, combine to one sequence: Let W = K × V. · · · − → Λn−2(K)

dn−2◦S−1

n−2◦dn−2

− − − − − − − − − − → Λn−1(V) (

Sn−1 dn−1)

− − − − → Λn(W) → 0 After proper identifications, (n = 2), this is elasticity sequence C ∞(K) J − → C ∞(M) (skw

div)

− − − → C ∞(W) → 0.

Richard S. Falk Finite Element Methods for Linear Elasticity

Approximation using Hilbert complex theory ?

Let W = K × V. Elasticity complex is: · · · − → Λn−2(K)

dn−2◦S−1

n−2◦dn−2

− − − − − − − − − − → Λn−1(V) (

Sn−1 dn−1)

− − − − → Λn(W) → 0 Can find finite element subspaces of Λn−2(K), Λn−1(V), and Λn(W) with bounded projections. However, operator S−1

n−2 does not make sense on finite element

spaces, since spaces not of same dimension and Sn−1 does not map to correct finite element space. So cannot obtain a finite element subcomplex and hence cannot apply our theory directly from this formulation.

Richard S. Falk Finite Element Methods for Linear Elasticity

Go back one step

· · · − → Λn−2(K) dn−2 − − − → Λn−1(K) dn−1 − − − → Λn(K) → 0 ր Sn−2 ր Sn−1 · · · − → Λn−2(V) dn−2 − − − → Λn−1(V) dn−1 − − − → Λn(V) → 0. · · · − → Λn−2

h

(K) dn−2 − − − → Λn−1

h

(K) dn−1 − − − → Λn

h(K) → 0

ր Sn−2,h ր Sn−1,h · · · − → Λn−2

h

(V) dn−2 − − − → Λn−1

h

(V) dn−1 − − − → Λn

h(V) → 0.

Each discrete complex is subcomplex of corresponding continuous

• complex. Only operators Sn−1 and Sn−2 need to be approximated.

Richard S. Falk Finite Element Methods for Linear Elasticity

Stability of continuous problem

To establish stability for continuous problem, prove inf-sup: Theorem: Given (ω, µ) ∈ L2Λn(Ω; K) × L2Λn(Ω; V), there exists σ ∈ HΛn−1(Ω; V) such that dn−1σ = µ, −Sn−1σ = ω. Moreover, we may choose σ so that σHΛ ≤ c(ω + µ), for a fixed constant c, where σ2

HΛ = σ2 + dn−1σ2.

Richard S. Falk Finite Element Methods for Linear Elasticity

Outline of key part of proof

Look for σ of form: σ = dn−2̺ + η ∈ HΛn−1(Ω; V) . By standard result, Can find η ∈ H1Λn−1(Ω; V) satisfying dn−1η = µ, and then τ ∈ H1Λn−1(Ω; K) satisfying dn−1τ = ω + Sn−1η. Since Sn−2 isomorphism from H1Λn−2(Ω; V) onto H1Λn−1(Ω; K), have ̺ ∈ H1Λn−2(Ω; V) with Sn−2̺ = τ. Then dn−1σ = dn−1dn−2ρ + dn−1η = µ and − Sn−1σ = −Sn−1dn−2̺ − Sn−1η = dn−1Sn−2̺ − Sn−1η = dn−1τ − Sn−1η = ω.

Richard S. Falk Finite Element Methods for Linear Elasticity

Remarks about proof

(i) Although elasticity problem only involves 3 spaces HΛn−1(Ω; V), L2Λn(Ω; V), and L2Λn(Ω; K), proof brings in 2 additional spaces: HΛn−2(Ω; V) and HΛn−1(Ω; K). (ii) Although Sn−1 is only S operator arising in formulation, Sn−2 plays key role in proof. (dn−1Sn−2 = −Sn−1dn−2). (iii) Do not fully use fact that Sn−2 is an isomorphism from Λn−2(Ω; V) to Λn−1(Ω; K), only that it is a surjection. (iv) Other slightly weaker conditions can be used in some places in the proof: (needed for some choices of stable finite element spaces).

Richard S. Falk Finite Element Methods for Linear Elasticity

Try to do analogous proof in discrete case

Let Λk

h be discrete k–forms: ω(x) = I fI(x)dxI, where {fI} are

piecewise polynomial functions with values in R. Assume following discrete complex is exact 0 − → Λ0

h d0

− → Λ1

h d1

− → Λ2

h → 0

and that projection operators Πh onto Λk

h are such that following

diagram commutes 0 − − → Λ0

d

− − → Λ1

d

− − → Λ2 − − → 0   Πh   Πh   Πh 0 − − → Λ0

h d

− − → Λ1

h d

− − → Λ2

h −

− → 0 Λk

h: use standard FE spaces giving subcomplex of de Rham.

Richard S. Falk Finite Element Methods for Linear Elasticity

What is needed?

Define discrete version of operators Kk and Sk: Kk,h = ΠhKk , Sk,h = ΠhSk. Problem in using same stability proof on discrete level: can’t expect Sn−2,h to be an isomorphism from Λn−2

h

(V) to Λn−1

h

(K). However, looking at proof, only need: (A) The operator Sn−2,h : Λn−2

h

(V) → Λn−1

h

(K) is onto.

Richard S. Falk Finite Element Methods for Linear Elasticity

Choosing finite element spaces to get stable discretization

Need to find combinations of discrete de Rham sequences for which Assumption (A) (Sn−2,h : Λn−2

h

(V) → Λn−1

h

(K) onto) is satisfied. Let n = 2. Given ω1 = −f1 f1

• dx1 +

−f2 f2

• dx2 ∈ Λ1

h(K),

find ω0 = (g1, g2)T ∈ Λ0

h(V) such that

S0,hω0 = πhS0ω0 = πh −g2 g2

• dx1 −

−g1 g1

• dx2
• =

−f1 f1

• dx1 +

−f2 f2

• dx2.

Can be reduced to checking degrees of freedom of discrete spaces.

Richard S. Falk Finite Element Methods for Linear Elasticity

Simple stable choice

P1Λ0

h(K) d0

− → P−

1 Λ1 h(K) d1

− → P0Λ2

h(K) → 0

P2Λ0

h(V) d0

− → P1Λ1

h(V) d1

− → P0Λ2

h(V) → 0.

✁ ✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆

• ✁

✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆ ❍ ❍ ❨ ✟ ✟ ✯ ❄ ✁ ✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆

• d0

✲

d1

✲ ✁ ✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆

• ✁

✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆ ❍ ❍ ❨ ❍ ❍ ❨ ✟ ✟ ✯ ✟ ✟ ✯ ❄❄ ✁ ✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆

• d0

✲

d1

✲

Top sequence: continuous P1, RT0, piecewise constants. Bottom sequence: continuous P2, BDM1, piecewise constants.

Richard S. Falk Finite Element Methods for Linear Elasticity

A simpler element

✁ ✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆

• ✁

✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆ ❍ ❍ ❨ ✟ ✟ ✯ ❄ ✁ ✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆

• d0

✲

d1

✲ ✁ ✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆

• ✁

✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆ ❍ ❍ ❨ ❍ ❍ ❨ ✟ ✟ ✯ ✟ ✟ ✯ ❄❄ ✁ ✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆

• d0

✲

d1

✲

Don’t use all the DOF in P2Λ0

h(V) to map onto P− 1 Λ1 h(K). Need

• nly P1Λ0

h(V) + 3 edge bubbles. Leads to reduced stress space and

same displacement space with same accuracy of approximation. Analogous to stable Stokes element P1 + edge bubbles – P0.

Richard S. Falk Finite Element Methods for Linear Elasticity

Remarks on 3-D elements

For r ≥ 0, Use exact sequences: Pr+1Λ0

h(K) d0

− → P−

r+1Λ1 h(K) d1

− → P−

r+1Λ2 h(K) d2

− → PrΛ3

h(K) → 0,

Pr+2Λ0

h(V) d0

− → P−

r+2Λ1 h(V) d1

− → Pr+1Λ2

h(V) d2

− → PrΛ3

h(V) → 0.

First sequence is usual sequence for N´ edel´ ec elements of first kind. Note second sequence not usual sequence for the Pr spaces (second kind N´ edel´ ec elements). Pr+3Λ0

h(V) d0

− → Pr+2Λ1

h(V) d1

− → Pr+1Λ2

h(V) d2

− → PrΛ3

h(V) → 0.

From FEEC, we know there are 2n−1 complexes in n dimensions.

Richard S. Falk Finite Element Methods for Linear Elasticity

Family of stable finite elements

Mixed elasticity with weakly imposed symmetry: Find (σ, u, p) ∈ Σh × Vh × Qh ⊂ H(div, Ω, M) × L2(Ω, V) × L2(Ω, K) such that (Aσ, τ) + (div τ, u) + (τ, p) = 0, τ ∈ Σh, (div σ, v) = (f , v), v ∈ Vh, (σ, q) = 0, q ∈ Qh. A family of elements: r ≥ 0, for n = 2 and n = 3:

• Σh ∼

= Pr+1Λn−1

h

(V)

• Vh ∼

= PrΛn

h(V)

• Qh ∼

= PrΛn

h(K)

Richard S. Falk Finite Element Methods for Linear Elasticity

Error estimates

Theorem: Suppose (σ, u, p) solves elasticity system and (σh, uh, ph) solves discrete elasticity system. Under reasonable hypotheses, and for appropriate projection operators Πh, σ − σh + p − ph ≤ C(σ − Πn−1

h

σ + p − Πn

hp),

u − uh ≤ C(σ − Πn−1

h

σ + p − Πn

hp + u − Πn hu),

dn−1(σ − σh) = dn−1σ − Πn

hdn−1σ.

For family just discussed, and 1 ≤ k ≤ r + 1, σ − σh + p − ph + u − uh ≤ Chk(σk + pk + uk).

Richard S. Falk Finite Element Methods for Linear Elasticity

Connections to Earlier Work

Most early work on reduced symmetry elements (e.g., PEERS, Fortin, Morley, Stenberg) based on use of stable pair of Stokes elements. Fits this framework with one basic modification, i.e., need to insert L2 projection (Πn

h) into top discrete exact sequence.

Λ0

h(K) d0

− → Λ1

h(K) Πn

h d1

− − − → Λ2

h(K) → 0

Λ0

h(V) d0

− → Λ1

h(V) d1

− → Λ2

h(V) → 0.

Spaces Λ1

h(K) and Λ2 h(K) correspond to velocity and pressure space

for Stokes.

Richard S. Falk Finite Element Methods for Linear Elasticity

PEERS (Arnold-Brezzi-Douglas)

For n = 2, letting B3 denote cubic bubble function, choose Λ1

h(V) = P− 1 Λ1(Th; V) + dB3Λ0(Th; V),

Λ2

h(V) = P0Λ2(Th; V),

Λ2

h(K) = P1Λ2(Th; K) ∩ H1Λ2(K),

Choose two remaining spaces: Λ0

h(V) = (P1 + B3)Λ0(Th; V),

Λ1

h(K) = S0Λ0 h(V).

✁ ✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆

• (Stokes mini-element)

✁ ✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆

• Πhd1

✲ ✁ ✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆

• ✁

✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆ ❍ ❍ ❨ ✟ ✟ ✯ ❄

• ✲

d0

✁ ✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆

• d1

✲

Known error estimate: σ − σh0 + p − ph0 + u − uh0 ≤ Ch(σ1 + p1 + u1).

Richard S. Falk Finite Element Methods for Linear Elasticity

Farhoul-Fortin method – improved stress approximation

Choose Λ1

h(V) = P1Λ1(Th; V),

Λ2

h(V) = P0Λ2(Th; V),

Λ2

h(K) = P1Λ2(Th; K) ∩ H1Λ2(K),

and two remaining spaces as Λ0

h(V) = P2Λ0(Th; V),

Λ1

h(K) = S0Λ0 h(V) ≡ P2Λ1(Th; K) ∩ H1.

✁ ✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆

• (Taylor-Hood)

✁ ✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆

• ✲

Πhd1

✁ ✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆

• ✁

✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆ ❍ ❍ ❨ ❍ ❍ ❨ ✟ ✟ ✯ ✟ ✟ ✯ ❄ ❄ ✲

d0

✁ ✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆

• ✲

d1

Richard S. Falk Finite Element Methods for Linear Elasticity

Some recent papers on reduced symmetry elements

• D. Boffi, F. Brezzi, M. Fortin, Reduced symmetry elements in linear

elasticity, Commun. Pure Appl. Anal. 8 (2009), no. 1, 95-121.

• B. Cockburn, J. Gopalakrishnan and J. Guzm´

an, A new elasticity element made for enforcing weak stress symmetry, Math. Comp., 79 (2010), 1331-1349.

• J. Gopalakrishnan and J. Guzm´

an, A second elasticity element using the matrix bubble, IMA J. Numer. Anal., 32 (2012), 352-272.

• J. Guzm´

an, A unified analysis of several mixed methods for elasticity with weak stress symmetry, J. Sci. Comp., 44 (2010), 156-169.

• R. Falk, Finite Elements for Linear Elasticity, in Mixed Finite

Elements, Compatibility Conditions, and Applications, Lecture Notes in Mathematics, Springer-Verlag, 1939 (2008), pp. 160-194.

Richard S. Falk Finite Element Methods for Linear Elasticity

Nonconforming elements with symmetric stresses

• D. N. Arnold and R. Winther, Nonconforming mixed elements for

elasticity, Dedicated to Jim Douglas, Jr. on the occasion of his 75th birthday. Math. Models Methods Appl. Sci. 13 (2003), no. 3, 295-307.

• J. Gopalakrishnan and J. Guzm´

an, Symmetric non-conforming mixed finite elements for linear elasticity, SIAM J. Numer. Anal., 49 (2011), no. 4, 1504-1520.

Richard S. Falk Finite Element Methods for Linear Elasticity

Methods using rectangular elements

• D. N. Arnold and G. Awanou, Rectangular mixed finite elements

for elasticity, Math. Models Methods Appl. Sci. 15 (2005), no. 9, 1417-1429.

• G. Awanou, A rotated nonconforming rectangular mixed element

for elasticity, Calcolo 46 (2009), no. 1, 49-60.

• G. Awanou, Two remarks on rectangular mixed finite elements for

elasticity, to appear in J. of Scientific Computing.

• G. Awanou, Rectangular Mixed Elements for Elasticity with

Weakly Imposed Symmetry Condition, to appear in Advances in Computational Mathematics.

Richard S. Falk Finite Element Methods for Linear Elasticity

Summary I

• While not fully fitting FEEC theory for Hilbert complexes when
• riginally developed, use of full commuting diagrams for whole

elasticity sequence with strong symmetry enabled construction of first stable pair of elements using polynomial shape functions (A-W). 0 − − → C 2(R)

J

− − → C 0(S)

div

− − → L2(R2) − − → 0   I 2

h

  I d

h

  I 0

h

0 − − → Qh

J

− − → Σh

div

− − → Vh − − → 0

Richard S. Falk Finite Element Methods for Linear Elasticity

Summary II

• BGG construction of elasticity sequence (with weak symmetry)

from de Rham sequence gave new understanding of structure and played key role in construction of new stable elements for elasticity (A-F-W). · · · − → Λn−1(K) dn−1 − − − → Λn(K) → 0 ր Sn−2 ր Sn−1 · · · − → Λn−2(V) dn−2 − − − → Λn−1(V) dn−1 − − − → Λn(V) → 0. Key identity: dn−1Sn−2 = −Sn−1dn−2 all dimensions

• By slightly modifying approach, old methods fit new framework.

Richard S. Falk Finite Element Methods for Linear Elasticity