Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 155

This is the weak equation for the relaxed minimum problem
1
2

[ P h ∇ v h ] 2 dx − fv h → min

v h ∈ M h

. ( 5 . 16 )

Only the part of the gradient that is projected onto X h contributes to the energy in
the variational formulation. The amount of the softening is fixed by the choice of
the target space of the projection.

∇ M h

E h

X h
Fig. 36. Projection of the gradient onto X h in the mixed method and the EAS
method, resp.
There is another characterization. The variational equations ( 5 . 2 ) h can be
rewritten in a form which leads to linear equations with a positive definite matrix.
We may choose a subspace E h of the L 2 -orthogonal complement of X h such that
∇ M h ⊂ X h ⊕ E h . ( 5 . 17 )
5.8 Remark. The mixed method ( 5 . 2 ) h is equivalent to the variational formulation
( ∇ u h , ∇ v) 0 , + (ε h , ∇ v) 0 , = (f, v) 0 ,

for all v ∈ M h ,

( 5 . 18 )

for all η ∈ E h , ( ∇ u h , η) 0 , + (ε h ,η) 0 , = 0

if the space E h of enhanced gradients satisfies the decomposition rule (5.17). Here
the relaxation of the variational form and the projector P h are defined by E h , i.e.,
by the orthogonal complement of the target space.
The proof of the equivalence follows Yeo and Lee [1996]. Let σ h ,u h be a
solution of ( 5 . 2 ) h . From (5.17) we have a decomposition
∇ u h = σ h − ε h

ε h ∈ E h .

with σ h ∈ X h and

From the first equation in ( 5 . 2 ) h we conclude that ∇ u h − σ h is orthogonal to

X h , and the uniqueness of the decomposition implies

σ h = σ h . When we insert
σ h =∇ u h + ε h in ( 5 . 2 ) h , we get the first equation of the system (5.18). The second
one is a reformulation of

X h and X h ⊥ E h .
The converse follows from the uniqueness of the solutions. The uniqueness
of the solution of (5.18) follows from ellipticity which in turn is given by Problem
5.10.

∇

u h +˜ ε h ∈

We note that in structural mechanics an equivalent concept was derived by
Simo and Rifai [1990] and called the method of enhanced assumed strains (EAS
method) .
The stability of the mixed method can be stated in terms of the enhanced
elements; cf. Braess [1998]. It also shows that the stability is not independent of
the choice of the space E h .