(Stationary penalized problem) For a
bounded set
with smooth boundary and given functions
and
find a function
satisfying the
relationships
where the operation
is given by the definition (
Bilinear form B
).

(Galerkin
approximation of stationary problem) We construct finite dimensional spaces
as follows.

Let
.
We take the increasing set of linearly independent functions
with the
properties
where we
introduced
We pose the problem of finding a function
such
that

We look for
and set consecutively
in the problem
(
Galerkin
approximation of stationary problem
). We end up with a system of
equations
The condition (
Assumption of coercivity
1
) implies that
is an invertible matrix and there is estimate
where the constant
is not dependent on
.
Let
then
We introduce the change of variable
:
Note that the operation
has the following
properties
for some constant
independent of
.
Therefore, there is a value of the constant
that makes the
operation
a contraction. By the proposition
(
Banach fixed point theorem
), for
such
there is a fixed point
:
Suppose
is such that the proposition (
Banach
fixed point theorem
) is applicable so
that
We aim to construct
for a greater constant
We subtract the above equalities and
obtain
We make the change of variable
:
The term
is a contraction with respect to
if
is chosen so that
with
taken from
.
The term
is a contraction if
is small enough. Hence, there is a solution
according to the proposition (
Banach
fixed point theorem
). We thus recover the value
for all
.

(2) According to the relationship
,
the last estimate and the proposition
(
Energy estimates for the
bilinear form
B
)
We calculate
and note that
is nonnegative and
thus
is nonpositive. Hence
and
according to
Thus
or