Finite Elements in Solid Mechanics - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 301

the pairing (3.14) is more appropriate. Find σ ∈ H( div ,)and u ∈ L 2 () 3 with

( C − 1 σ, τ) 0 + ( div τ,u) 0 =

0 for all τ ∈ H( div , ), τ n = 0on 1 ,

for all v ∈ L 2 () 3 , ( div σ, v) 0 =− (f, v) 0

= g

σn on 1 .

( 3 . 22 )
We assume that an inhomogeneous boundary condition has been reduced to a
homogeneous one in the sense of Ch. II, §2. The equations (3.22) are the Euler
equations for the saddle point problem

( C − 1 σ, σ) 0 −→

min
σ ∈ H( div ,)

!

with the restriction

div σ = f

and the boundary condition σn = g on 1 . This is often called the dual mixed
method .
Just as in H 0 () where boundary values for the function are prescribed, in
the (less regular) space H( div ,) we can specify the normal components on the
boundary. This becomes clear from the jump conditions in Problem II.5.14. Here
we assume that the boundary is piecewise smooth.
Although in (3.22) formally we required only that u ∈ L 2 () 3 , in fact the
solution satisfies u H 1 = C − 1 σ ∈ () . It follows from (3.22) that ε(u) ∈ L 2 () .

Indeed, suppose i, j ∈{ and that only τ ij = τ ji are nonzero. In addition,

let τ ij ∈ C 0 () . Then writing w instead of τ ij , it follows from (3.22) that

1 , 2 , 3 }

u i ∂w dx

1
2

u j ∂w
∂x i

C − 1 σ) ij wdx.

∂x j +

=− (

(s) u) ij exists in
the weak sense, and coincides with ( C − 1 σ) ij ∈ L 2 () . Now Korn's first inequality
implies u ∈ H 1 () 3 . Finally, we apply Green's formula. Because of the symmetry,
it follows that for all test functions τ as in (3.22),

Recalling Definition II.1.1, we see that the symmetric gradient ( ∇

u · τnds =

∇ u : τdx +

u · div τdx

∂

(s) u : τdx +

=

∇

u · div τdx

C − 1 σ : τdx

= + u · div τdx = 0 .

Since this holds for all test functions, it follows that u = 0on 0 = ∂ \ 1 .