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112 3. Finite Element Methods for Linear Elliptic Problems

• the smoothness of the boundary of the domain,
• the shape of the domain,
• thesmoothness of thecoefficients and theright-hand sideof the
differential equation and the boundary conditions,
• the kind of the transition of boundary conditions in those points,
where the type is changing,

which can be quite restrictive as a whole. Therefore, in what follows we
often assume only the required smoothness. Here we cite as an example
one regularity result ([13, Theorem 8.12]).

2
Theorem 3.18 Suppose Ω is a bounded C -domain and Γ 3 = ∂Ω.Further,
1 ¯
2
∞
assume that k ij ∈ C (Ω),c i ,r ∈ L (Ω) ,f ∈ L (Ω) ,i, j ∈{1,... ,d},
2
as well as (3.16). Suppose there exists some function w ∈ H (Ω) with
γ 0 (w)= g 3 on Γ 3 .Let ˜u = u + w and let u be a solution of (3.35).Then
2
˜ u ∈ H (Ω) and
˜u 2 ≤ C{ u 0 + f 0 + w 2 }
with a constant C> 0 independent of u, f,and w.
One drawback of the above result is that it excludes polyhedral domains.
If the convexity of Ω is additionally assumed, then it can be transferred
to this case. Simple examples of boundary value problems in domains with
reentrant corners show that one cannot avoid such additional assumptions
(see Exercise 3.5).

Exercises

3.5 Consider the boundary value problem (1.1), (1.2) for f =0 in the

sector Ω := (x, y) ∈ R 2
x = r cos ϕ, y = r sin ϕ with 0 <r < 1, 0 <ϕ<

α for some 0 <α < 2π, thus with the interior angle α. Derive as in (1.23),
by using the ansatz w(z):= z 1/α ,a solution u(x, y)= w(x + iy) for an
appropriate boundary function g. Then check the regularity of u,thatis,
k
u ∈ H (Ω), in dependence of α.
3.6 Consider the problem (1.29) with the transmission condition (1.30)
and, for example, Dirichlet boundary conditions and derive a variational
formulation for this.

3.7 Consider the variational formulation:
1
Find u ∈ H (Ω) such that

1
∇u ·∇vdx = fv dx + gv dσ for all v ∈ H (Ω) , (3.37)
Ω Ω ∂Ω