Finite Elements in Solid Mechanics - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 336

Here τ = τ(x) is defined (almost everywhere) on ∂ as the direction of the tangent
in the counterclockwise direction. We endow the space (6.3) with the norm

1
2 .

2
0

2

η 0 , rot : = ( η + rot η

0 )

( 6 . 4 )

In terms of the general theory in Ch. III, §4, the bilinear form b for the mixed
formulation is given by b(w, θ, η) : = ( ∇ w − θ,η) 0 . Now we specialize η to be

an element of L 2 () 2

of the form η = curl p . Then because of the orthogonality

of the rotation and the gradient,

b(w, θ ; η) = ( ∇ w − θ,η) 0 = 0 − (θ, η) 0 ≤ θ 1 η − 1 ,

H 0 (), θ H 0 () 2 , and thus for w ∈ ∈

b(w, θ, η)
w 1 + θ 1 ≤ η − 1 .

sup
w,θ

In order to ensure that the inf-sup condition holds, we have to endow M with a
norm which is weaker than the L 2 -norm, that is with the one which is dual to (6.4),
i.e.,

· H − 1 ( div ,) ; see below.

As shown in Brezzi and Fortin [1986], the analyis is simplified if we use the
Helmholtz decomposition of the shear term into a gradient field and a rotational
field. Using the decomposition we get expressions and estimates which involve
the usual Sobolev norms.

The Helmholtz Decomposition

In the following we shall see that the space

H − 1 ( div ,) : ={ η ∈ H − 1 () 2 div η ∈ H − 1 () }

;

with the graph norm (5.17) is the dual space of H 0 ( rot ,) . Clearly,

H 0 ( rot ,) ⊂ L 2 () 2 ⊂ H − 1 ( div , ).

As usual, we identify functions in L 2 ()/ R which differ only by a constant. The

norm of an element in this space is just the L 2 -norm of the representer which is
normalized to have zero integral; see Problem III.6.6.