Convergence of finite elements for generic parabolic operator.

roblem

(Dirichlet
problem for generic parabolic operator) Find the function
,
,
such
that
for some functions
and
.
The
is assumed to be a bounded domain with a
boundary. The functions
and
are assumed to satisfy the compatibility conditions of the proposition
(
Parabolic regularity 2
). The form of
the operator
is given by the formula (
Operator L 2
) and the
coefficients of
satisfy the definition (
Elliptic
differential operator
).

Problem

(Variational Dirichlet problem for generic parabolic operator)
Find
such
that
for any
.
The form of
is given by the definition (
Bilinear form B
2
).

Problem

(Time dependent elliptic problem)
Under assumptions of the definition
(
Dirichlet
problem for generic parabolic operator
) find the function
such
that
We introduce the notation
.
We choose the number
so that the estimate

(Partially inverted problem) We assume
existence of approximate solution
for the problem
(
Solution operator
for generic elliptic problem
). Then we introduce the
problem
and consider the function
,
to be the approximation of
.

Condition

(Properties of solution operator
2) We assume that the operator
has the following properties

We proceed to prove the condition
(
Properties of solution operator
2
)-2. We introduce the notation
.
According to the formula (
Energy estimate
2
)
Note that
hence
and by the proposition
(
Energy estimates for the
bilinear form
B
)
Therefore, for any
and according to the condition
(
Finite dimensional
approximation 1
) this
implies
We now
estimate
We introduce the function
such that
then
for any
.
We apply the proposition
(
Energy estimates for the
bilinear form
B
):
and substitute the recent result
:
We apply the condition (
Finite
dimensional approximation 1
) with
to the term
where we have the freedom of choosing
:
We invoke the proposition (
Boundary
elliptic regularity
) for
with
:
:
Thus
The results
taken with the proposition (
Boundary
elliptic regularity
) applied to
:
constitute
It remains to prove the
part of the condition
(
Properties of solution operator
2
)-2. We introduce the notation
for the time derivative of the form
so that the differentiation of
is written as
According to the formula (
Energy estimate 2
)
where we introduced an arbitrary
.
Thus,
may be regarded as
.
Each term has the following energy
estimates
We
derive
We have the inequality of the form
.
We
introduce
and we are given that the domain of
is such that
.
In this context
and it must lie between the roots of the quadratic polynomial for permitted
values of
.
From this analysis there is constant
such
that
Hence, we
continue
and apply the result
and the condition (
Finite
dimensional approximation 1
)
The
is a solution of the problem
Hence by the proposition
(
Energy estimates for the
bilinear form B
) and (
Boundary
elliptic regularity
)
.
We
conclude
which is not quite the goal yet. To gain the extra power of
we perform the following calculation.

Let
is the adjoint of
and for arbitrary
let
is such
that
We act as above:
We integrate by parts in the last
term
We apply the condition (
Finite
dimensional approximation
1
)
We apply the proposition (
Boundary
elliptic regularity
)
and the results
and arrive to the remaining statement
(2):

We introduce the
quantity
and calculate the error equation exactly as in the proof of the proposition
(
Galerkin convergence 4
), (change
):
We apply the proposition (
Partial
inversion
lemma
):
and estimate every term on the right.