Finite Elements in Solid Mechanics - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 303

holds with c from Korn's inequality on the subspace

(s) v, τ) 0

V ={ (η, v) ∈ X ;− (η, τ ) 0 + ( ∇ = 0 for τ ∈ M } .

Thus the bilinear form a is V -elliptic.
The inf-sup condition is easily verified. We need only evaluate b with η = τ
and v

0.
As a second possibility, using the same bilinear form a , we can work with
the pairing

=

= L 2 () × L 2 () 3 ,M : ={ τ ∈ H( div ,) ; τn =

0on 1 } ,

X :

( 3 . 25 )

b(ε, u ; τ) =− (ε, τ ) 0 − (u, div τ) 0 .

The argument is the same as in the second formulation of the Hellinger-Reissner
principle.
In regard to the finite element approximation, we should mention one differ-
ence as compared with the Stokes problem. The bilinear form a is elliptic on the
entire space X only for the first version of the Hellinger-Reissner principle, while
in the other cases it is only V -elliptic. The ellipticity on V h can only be obtained
if the space X h is not too large in comparison with M h ; see Problem III.4.18. On
the other hand, since the inf-sup condition requires X h to be sufficiently large, the
finite element spaces X h and M h have to fit together.
There is one more reason why it is not easy to provide stable, genuine ele-
ments for the Hu-Washizu principle. Here elements are said to be genuine if they
are not equivalent to some elements for the Hellinger-Reissner theory or for the
displacement formulation.

3.8 First Limit Principle of Stolarski and Belytschko [1966]. Assume that
u h ∈ V h ,ε h ∈ E h , and σ h ∈ S h constitute the finite element solution of a problem
by the Hu-Washizu method. If the finite element spaces satisfy the relation

S h ⊂ C E h , ( 3 . 26 )

then (σ h ,u h ) is the finite element solution of the Hellinger-Reissner formulation
with the (same) spaces S h and V h .
Proof. The arguments in the proof are purely algebraic and apply to the pairings
(3.24) and (3.14), or (3.25) and (3.15), respectively. In order to be specific we
restrict ourselves to the first case and assume that

(ε h , C η) 0 − (η, σ h ) 0 = 0 for all η ∈ E h ,

(s) v, σ h ) 0 = (f, v) 0 −

( ∇ g · vdx for all v ∈ V h ,

1

(s) u h ,τ) 0

− (ε h ,τ) 0 + ( ∇ = 0 for all τ ∈ S h .

( 3 . 27 )