Finite Elements in Solid Mechanics - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 340

is a so-called reduction operator , i.e., a linear mapping defined on the finite element
space for the shear terms which does not affect the elements in h .
If possible, the finite element calculations are performed using the displace-
ment model (6.13), since it leads to systems of equations with positive definite
matrices and with fewer unknowns. On the other hand, for the convergence anal-
ysis, it is still best to use the mixed formulations. However, there is a problem: In
general, the functions in h cannot be represented in the form

γ h = grad r h + curl p h ,

where r h and p h again belong to finite element spaces. Thus, except for a special
case treated by Arnold and Falk [1989], a modification of the Helmholtz decom-
position is necessary.
The notation for the following finite element spaces is suggested by the vari-
ables in (6.11).

6.3 The Axioms of Brezzi, Bathe, and Fortin [1989]. Suppose the spaces

W h ⊂ H 0 (), h ⊂ H 0 () 2 ,Q h ⊂ L 2 ()/ R , h ⊂ H 0 ( rot ,)

and the mapping R h defined in (6.14) have the following properties:
(P 1 ) ∇ W h ⊂ h , i.e., the discrete shear term γ h :

= t − 2 ( ∇ w h − R h θ h ) lies in h .
(P 2 ) rot h ⊂ Q h - this requirement is consistent with γ h ∈ H( rot ,) , and thus
with rot γ h ∈ L 2 () .
(P 3 ) The pair ( h ,Q h ) satisfies the inf-sup condition

( rot ψ h ,q h )
ψ h 1 q h 0 =

inf
q h ∈ Q h

sup
ψ h ∈ h

: β> 0 ,

where β is independent of h . The spaces are thus suitable for the Stokes

problem.
(P 4 ) Let P h be the L 2 -projector onto Q h . Then

for all η ∈ H 0 () 2 , rot R h η = P h rot η

i.e., the following diagram is commutative:

rot
−−→ L 2 ()

H 0 () 2

R h 4

4

P h

rot
−−→ Q h .

h