Spacial discretization for Heat equation with
Dirichlet boundary conditions.

roblem

(Galerkin approximation 2) We define the
finite dimensional approximation to the solution of the problem
(
Heat equation weak formulation
1
) as the solution
of the
problem
where the
is some
-based
approximation of
.

(1) We introduce the notations
so
that
We estimate, according to the proposition
(
Ritz projection convergence
1
),
We put the
over the relationship
and
conclude
To estimate the
we
write
and substitute the definition of
:
We want to remove everything involving
.
Hence we substitute the relationships
and
We substitute the relationship
:
or
We have
,
hence, we substitute
in the last relationship and
obtain
Consequently
and we continue deriving the
consequences:
According to the initial conditions for
and
,
and we use the proposition (
Ritz
projection convergence 1
) in the second
term:
The statement now follows from the obtained
results

Proof

(2) We
have
According to the proposition (
Ritz
projection convergence
1
)
We return to the relationship
from the first part of this proof and substitute
:
We substitute
and
(proposition (
Cauchy
inequality
)):
and we continue deriving
consequences:
and the second estimate follows after application of the proposition
(
Ritz projection convergence 1
)
to the term
.