Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 184

9.3 Algorithm.
set σ 1 ,r = 0;
for i =

1 , 2 ,... , until an entire circuit around z is completed
{ fix σ i,l

such that σ ω z · nds = fψ z dx − σ ω z · nds ;

e i T i e i − 1

1

fix σ i + 1 ,r such that [[ σ ω z · n ]] =−

2 [[

∇ u h · n ]] o n e i ;

}
It follows from Lemma 9.2 that the old and the new value of the normal component
σ 1 ,r coincide when an entire circuit around z has been completed.
If z is a node on ∂ , the construction has to be modified in an obvious way.
Here ∂ω z shares two edges with ∂ . We start at one of them and proceed as in
Algorithm 9.3 until we get to the other edge on ∂ . There is no problem since we
do not return to the edge of departure.

9.4 Theorem. Let u h be the finite element solution with P 1 elements and

σ = σ ω z

z

where the functions σ ω z are constructed by Algorithm 9.3. Then

∇ (u − u h ) 0 ≤ σ

0 + ch f − f h 0 . ( 9 . 8 )

Proof. Each edge has two nodes, and each triangle has three nodes. Moreover, we

have z ψ z (x) = 1 in each triangle T , and

fψ z dx = f 1 dx = f h dx.

T T T

z

Therefore it follows from (9.7) that the sum σ satisfies (9.4). The influence of the
data oscillation was discussed in §8, and the theorem of Prager and Synge yields
(9.8).
The theory that led to Theorem 9.4 differs from the theory in the previous
section. The estimator with local Neumann problems is also based on saddle point
problems, but an additional discretization is required. Nevertheless, the error esti-
mator σ

0 is comparable to the residual error estimator (8.11). Consequently,
the new estimator is also efficient.

9.5 Theorem. There is a constant c that depends only on the shape of the triangles
such that

σ ≤ cη R + ch f − f h ( 9 . 9 ) 0 0

≤ c | u − u h | 1 + ch f − f h 0 . ( 9 . 10 )