Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 152

5.6 Theorem. Let u h be the finite element solution with the P 1 element and σ h
be the solution of the mixed method with the Raviart-Thomas element on the same
mesh. Then

∇ u − σ h 0 ≤ c ∇ (u − u h ) 0 + ch inf
f h ∈ M

0 f − f h 0

with a constant c depending only on the shape regularity.
The proof is more involved and will be provided in §9 in connection with
a posteriori error estimates.
The error estimate for the u -component in (5.13) is weaker than that for
standard finite elements. On the other hand, the Raviart-Thomas element is more
robust than the standard method for a class of problems that we will encounter in
Ch. VI. Moreover the above disadvantage can be eliminated by a postprocessing
procedure that will be described briefly in the next subsection.
Since the mixed method with Raviart-Thomas elements is stable, we can
modify the diagram (5.12) such that the domain becomes H( div ) . We restrict
ourselves to k = 0 and define ρ as follows: Given σ ∈ H( div ) , let σ h = ρ σ ∈
RT 0 be the solution of the mixed method

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

0 .

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

Since the divergence operator is surjective, we have exact sequences in addition to
the commuting diagram property. Some larger diagrams play an important role in
the construction of modern finite element spaces; see Arnold, Falk, and Winther
[2006].

div
−−→ L 2 () −−→

H( div ,) 0

ρ 4

4

k

div
−−→ M

k

RT k −−→ 0 .

Implementation and Postprocessing

In principle, the discretization leads to an indefinite system of equations. It can be
turned into a positive definite system by a trick which was described by Arnold
and Brezzi [1985].
Instead of initially choosing the gradients to lie in a subspace of
H( div ,) , we first admit gradients in L 2 () 2 , and later explicitly require that
div σ h ∈ L 2 () . Equivalently, we require that the normal components σ h · n do
not have jumps on the edges. To achieve this, we enforce the continuity of σ h n on
the edges as an explicit constraint. This introduces a further Lagrange multiplier.