Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 90

s

N

Fig. 25. Reflection of a convex domain along the edge N on which a Neumann
condition is given

We see from Example 2.1 with a domain with reentrant corner that the as-
sumptions on the boundary cannot be dropped since the solution there is not in
H 2 () .
We now give an example to show that the situation is more complicated if a
Neumann condition is prescribed on a part of the boundary . Let be the convex
domain on the right-hand side of the y -axis shown in Fig. 25. Suppose the Neumann
condition

∂u
∂ν =

0

is prescribed on N : ={ (x, y) ∈ ∂ ; x = 0 } , and that a Dirichlet boundary

condition is prescribed on D : = \ N . The union of with its reflection in N

defines a symmetric domain s . Set

u( − x,y) = u(x, y) for (x, y) ∈ s \ .

Then its continuation is also a solution of a Dirichlet problem on s . But since
s has a reentrant corner, the solution is not always in H 2 ( s ) , which means that
u ∈ H 2 () cannot hold for all problems on with mixed boundary conditions.

Error Bounds in the Energy Norm

In the following, suppose that is a polygonal domain. This means that it can be
partitioned into triangles or quadrilaterals. In addition, in order to use Theorem 7.2,
suppose is convex.

7.3 Theorem. Suppose T h is a family of shape-regular triangulations of . Then

the finite element approximation u h ∈ S h = M

0 (k ≥

1 ) satisfies

u − u h 1 ≤ ch u 2
≤ ch f 0 .

( 7 . 3 )

Proof. By the convexity of , the problem is H 2 -regular, and

u 2 ≤ c 1 f 0 .By

Theorem 6.4, there exists v h ∈ S h with u − v h = u − v h ≤ c 2 h u

2 , . 1 , 1 ,h

Combining these facts with Cea's Lemma gives (7.3) with c : = ( 1 + c 1 )c 2 c 3 /α .