Backward Euler time discretization for Heat equation
with Dirichlet boundary conditions.

roblem

(Backward Euler problem) We introduce a
time step
,
mesh
and the time derivative approximation
.
We seek the array
of functions
that satisfies the conditions

We split the error term as
follows
We estimate the components
and
according to the procedure of the proof of the proposition
(
Galerkin convergence 2
)-1. The
has exactly the same
estimate

We estimate
as
follows:
We want to remove all the spacial
terms. We substitute the relationships
and
:
We substitute the relationship
taken at
:
where we introduced the
notation
We set
in the equality
and
obtain
hence
We substitute the definition of
:
and
obtain
thus
or
We apply the last inequality repeatedly and arrive to the
estimate
where the
is estimated as in the proof of the proposition
(
Galerkin convergence
2
)-1:
It remains to estimate the
:
We
have
and we apply the the proposition
(
Ritz projection convergence
1
):
Thus,
The estimation of
is done with similar
means: