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50 2. Finite Element Method for Poisson Equation

where

1 1 2
F(v):= a(v, v) − f, v 0 = v − f, v 0 .
a
2 2

Proof: (2.10) ⇒ (2.11):
Let u be a solution of (2.10) and let v ∈ V be chosen arbitrarily. We define
w := v − u ∈ V (because V is a vector space), i.e., v = u + w. Then, using
the bilinearity and symmetry, we have
1
F(v)= a(u + w, u + w) − f, u + w 0
2
1 1
= a(u, u)+ a(u, w)+ a(w, w) − f, u − f, w 0 (2.12)
0
2 2
1
= F(u)+ a(w, w) ≥ F(u) ,
2
where the last inequality follows from the positivity of a; i.e., (2.11) holds.
(2.10) ⇐ (2.11):
Let u be a solution of (2.11) and let v ∈ V , ε ∈ R be chosen arbitrarily. We
define g(ε):= F(u + εv)for ε ∈ R.Then
g(ε)= F(u + εv) ≥ F(u)= g(0) for all ε ∈ R ,
because u + εv ∈ V ; i.e., g has a global minimum at ε =0.
It follows analogously to (2.12):
1 ε 2
g(ε)= a(u, u) − f, u + ε (a(u, v) − f, v )+ a(v, v) .
0
0
2 2
Hence the function g is a quadratic polynomial in ε, and in particular,
1
g ∈ C (R) is valid. Therefore we obtain the necessary condition

0= g (ε)= a(u, v) − f, v
0
for the existence of a minimum at ε =0. Thus u solves (2.10), because
v ∈ V has been chosen arbitrarily.
For applications e.g. in structural mechanics as above, the minimization
problem is called the principle of minimal potential energy.

Remark 2.4 Lemma 2.3 holds for general vector spaces V if a is a sym-
metric, positive bilinear form and the right-hand side f, v is replaced by
0
b(v), where b : V → R is a linear mapping, a linear functional. Then the
variational equation reads as
find u ∈ V with a(u, v)= b(v) for all v ∈ V, (2.13)

and the minimization problem as

find u ∈ V with F(u)=min F(v) v ∈ V , (2.14)