Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 34

§ 2. Variational Formulation of Elliptic
Boundary-Value Problems of Second Order

A function which satisfies a given partial differential equation of second order and
assumes prescribed boundary values is called a classical solution provided it lies in
C 2 ()

C 1 ( ¯ )
in the case of Neumann boundary conditions, respectively. Classical solutions exist
if the boundary of the underlying domain is sufficiently smooth, and if certain
additional conditions are satisfied in the case where Neumann boundary conditions
are specified on part of the boundary. In general, higher derivatives of a classical
solution need not be bounded (see Example 2.1), and thus the simple convergence
theory presented in Ch. I for the finite difference method may not be applicable.
In this section we discuss the variational formulation of boundary-value prob-
lems. It provides a natural approach to their numerical treatment using finite ele-
ments, and also furnishes a simple way to establish the existence of so-called weak
solutions .

C 0 ( ¯ ) in the case of Dirichlet boundary conditions, and in C 2 () ∩ ∩

Fig. 7. Domain with reentrant corner (cf. Example 2.1)

2.1 Example. Consider the two-dimensional domain

2

; x 2 + y 2 < 1 ,x< 0or y> 0

={ (x, y) ∈ R } ( 2 . 1 )

2

= z 2 / 3 with reentrant corner (see Fig. 7) and identify R with C . Then w(z) : is

analytic in , and its imaginary part u(z) : = Im w(z) is a harmonic function

solving the boundary-value problem
u = 0 in ,

sin ( 2

3 π
2

u(e iϕ ) =

3 ϕ)

for 0 ≤ ϕ ≤ ,

u = 0 elsewhere on ∂.

Since w (z) =

2

3 z − 1 / 3 , even the first derivatives of u are not bounded as z →

0.
— The singularity will be no problem when we look for a solution in the right
Sobolev space.