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2.2. The Finite Element Method with Linear Elements 55

d-dimensional measure 0) and, in particular, u =0 on ∂Ω. This will be
described inSection3.1.

Exercises

2.1
(a) Consider the interval (−1, 1); prove that the function u(x)= |x| has

the generalized derivative u (x)=sign(x).
(b) Does sign(x) have a generalized derivative?
) N 2
2.2 Let Ω= Ω l ,N ∈ N, where the bounded subdomains Ω l ⊂ R
l=1
are pairwise disjoint and possess piecewise smooth boundaries. Show that
1
a function u ∈ C(Ω) with u| ∈ C (Ω l ), 1 ≤ l ≤ N, has a weak derivative
Ω l )
2
∂ i u ∈ L (Ω),i =1, 2, that coincides in N Ω l with the classical one.
l=1
2.3 Let V be the set of functions that are continuous and piecewise con-
tinuously differentiable on [0, 1] and that satisfy the additional conditions
u(0) = u(1) = 0. Show that there exist infinitely many elements in V that
minimize the functional
1
2 2
F(u):= 1 − [u (x)] dx.

0
2.2 The Finite Element Method
with Linear Elements

The weak formulation of the boundary value problem (2.1), (2.2) leads to
particular cases of the following general, here equivalent, problems:
Let V be a vector space, let a : V × V → R be a bilinear form, and let
b : V → R be a linear form.
Variational equation:

Find u ∈ V with a(u, v)= b(v) for all v ∈ V. (2.21)
Minimization problem:

Find u ∈ V with F(u)= min F(v) v ∈ V ,

1 (2.22)
where F(v)= a(v, v) − b(v) .
2
The discretization approach consists in the following procedure: Replace
V by a finite-dimensional subspace V h ; i.e., solve instead of (2.21) the finite-
dimensional variational equation,

find u h ∈ V h with a(u h ,v)= b(v) for all v ∈ V h . (2.23)