Proof
Let
be a solution of the problem
(
Partially
inverted semi discrete parabolic problem 1
) with
.
Then
and, for
being the solution operator of the problem
(
Partially
inverted semi discrete parabolic problem 1
),
It may be shown that there is an estimate for
similar to the one of the proposition
(
Instant
smoothness of solution operator for PP 1
),
hence
by the conditions of this proposition, the condition
(
Finite dimensional
approximation 1
) and proposition
(
Characterization of Htilde
spaces
)
Hence, it is enough to prove the statement for
.
We intend to apply the proposition
(
Partial inversion lemma 2
)3. We
introduce
and recall from the proof of the proposition
(
Galerkin convergence 4
)
that
Hence,
according to the condition
(
Properties of solution
operator
)2
and by the proposition
(
Instant
smoothness of solution operator for PP
1
)
Similarly,
Finally, for any
by symmetry of
as in the condition (
Properties of
solution
operator
)1,
because
.
Hence,
The statement now follows from the proposition
(
Partial inversion lemma 2
)3.
