Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 35

Variational Formulation

Before formulating linear elliptic problems as variational problems, we first present
the following abstract result.

2.2 Characterization Theorem. Let V be a linear space, and suppose

a : V × V → R

is a symmetric positive bilinear form, i.e., a(v, v) > 0 for all v ∈ V,v = 0 .In

addition, let

: V → R

be a linear functional. Then the quantity

1
2 a(v, v) − , v

J(v) : =

attains its minimum over V at u if and only if

a(u, v) = , v for all v ∈ V. ( 2 . 2 )

Moreover, there is at most one solution of (2.2).
Remark. The set of linear functionals is a linear space. Instead of (v) , we prefer
to write

in order to emphasize the symmetry with respect to and v .
Proof. For u, v ∈ V and t ∈ R

, v

,wehave

1
2 a(u + tv,u + tv) − , u + tv

J(u + tv) =

1

2 t 2 a(v, v).

= J (u) + t [ a(u, v) − , v ] + ( 2 . 3 )

If u ∈ V satifies (2.2), then (2.3) with t = 1 implies

1
2 a(v, v)

J(u + v) = J (u) + for all v ∈ V ( 2 . 4 )

> J (u), if v = 0.

Thus, u is a unique minimal point. Conversely, if J has a minimum at u , then for
every v ∈ V , the derivative of the function t → J(u + tv) must vanish at t = 0.

By (2.3) the derivative is a(u, v) − , v , and (2.2) follows.

The relation (2.4) which describes the size of J at a distance v from a minimal
point u will be used frequently below.