Finite Elements in Solid Mechanics - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 307

3.10 Lemma. Assume that (3.34) and (3.35) hold. Let the mapping div h : X h →
L 2 () satisfy

div v − div h v 0 ≤ ch v 2 ( 3 . 38 )

and

div h v = 0 div v = 0 . if ( 3 . 39 )

Then we have

div h u 0 ≤ c h f 0

λ div u − ( 3 . 40 )

where u denotes the solution of the variational problem (3.31).
Proof. By (3.34) and (3.37) there exists w ∈ H 2 () ∩ H 0 () such that div w =
div u and

div u 1 ≤ cλ − 1 w 2 ≤ c f 0 .

From (3.38) we conclude that

div h w 0 ≤ c h w 2 ≤ c hλ − 1

div w − f 0 . ( 3 . 41 )

Since div (w − u) = 0, it follows from (3.39) that div h (w − u) = 0 and div u −

div h u = div w − div h w . Combining this with (3.41) we obtain

div h w 0 ≤ c hλ − 1

div u − div h u 0 = div w − f 0 ,

and the proof is complete.
Now the discretization error for nearly incompressible material is estimated
by the lemma of Berger, Scott, and Strang [1972]. The term for the approximation
error is the crucial one. Let P h : H 2 () ∩ H 0 () → X h be an interpolation
operator. It is sufficient to make provision for
v − P h v 1 + div v − div h v 0 ≤ ch v 2 ,

div h v h 0 ≤ c v h 1 for all v h ∈ X h .
These inequalities are clear for the model problem in (3.35). Moreover, let the
mesh-dependent bilinear form a h be defined by polarization of the quadratic form
in (3.36). It follows from Lemma 3.10 that we have for w h ∈ X h
a h (u − P h u, w h ) = µ(ε(u − P h u), ε(w h )) 0 + λ( div u − div h u, div h w h ) 0 ,

≤ ch u 2 w h 1 + ch u 2 div h w h 0 ,

≤ ch u 2 w h 1 .
Finally, the consistency error is given by the term λ( div u −

div h u, div h w h ) 0 .An
estimate of this expression is already included in the formula above, and we have
a robust estimate with a constant c that does not depend on λ , namely

u − u h 1 ≤ ch f 0 .

The following theory describes why uniform convergence cannot be expected
with some discretizations. The equation (3.29) is included as a special case if we
set X : = H 1 () , a 0 (u, v) : div v , and t 2

= 2 µ(ε(u), ε(v)) 0 , Bv : = = 1 /λ .