Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 182

0 for all τ h ∈

RT 0 .
Thus the minimizing function σ h and the Lagrange multiplier w h are characterized
by the equations

with v being the Lagrange multiplier. Note that div τ h + f h ∈ M

(σ h ,τ) 0 + ( div τ,w h ) 0 = ( ∇ u h ,τ) 0 for all τ ∈ RT 0 ,

( 9 . 3 )

0 .

( div σ h ,v) 0 =− (f h ,v) 0 for all v ∈ M

By Green's formula we obtain ( ∇ u h ,τ) 0 =− (u h , div τ) 0 since the boundary terms
vanish. Let Q h be the L 2 projector onto

0 . Then (u h , div τ) 0 = (Q h u h , div τ) 0

M

for all τ ∈ RT 0 , and (9.3) can be rewritten

(σ h ,τ) 0 + ( div τ,w h + Q h u h ) 0 = 0 for all τ ∈ RT 0 ,

0 .

( div σ h ,v) 0 =− (f h ,v) 0 for all v ∈ M

The pair (σ h ,w h + Q h u h ) is a solution of the mixed method with the Raviart-
Thomas element.
Finally, we note that (9.3) 2 implies that div σ h =− f h holds in the strong
sense. Indeed, the expressions on both sides belong to

0 , and the relation (9.3) 2

M

is tested with functions in the same space.
Of course, the numerical solution of the mixed method is too expensive when
only an error estimate is desired. An approximation σ ∈ RT 0 will be sufficient and

will be constructed from u h by a simple postprocess.
We consider the space of broken Raviart-Thomas functions

τ ∈ L 2 () 2 a T

b T

+ c T x

,a T ,b T ,c T ∈ R

for T ∈ T h . RT − 1 : = ; τ | T =

y

The normal components are not required to be continuous, and RT 0 ∩

H( div ) . The degrees of freedom of the finite element functions in RT − 1 are the
normal components on the edges, but the values at the two sides of interior edges
may differ. Therefore, two degrees of freedom are associated to each inner edge;
see Fig. 45 below.
Note that

= RT − 1

∇ u h and σ h belong to RT − 1 . Moreover, in each triangle div ∇ u h =

0. (We do not consider div ∇ u h as a global function on .) We construct a suitable

σ by determining a suitable σ = σ −∇ u h . Find σ : ∈ RT − 1 such that

div σ

=− f h in each T ∈ T h ,

( 9 . 4 )

[[ σ · n ]] =− [[ ∇ u h · n ]] on each interior edge e.

Let z be a node of the triangulation. The construction will be performed on
patches

= { T ; z ∈ ∂T } .

ω z :

As a preparation we have