Crank-Nicolson time discretization for the Heat
equation with Dirichlet boundary conditions.

roblem

(Crank-Nicolson problem
for heat equation) We introduce a time step
,
mesh
the time derivative approximation
and the averaging operation
.
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 substitute the relationships
and
:
We substitute the relationship
taken at
:
where we introduced the
notation
We set
in the equality
and
obtain
hence
or, after substitution of definitions for
and
,
and after cancellation for
:
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
:
The
was estimated when proving the proposition
(
Backward Euler convergence
2
):
We estimate
using the proposition
(
Integral form of Taylor
decomposition
) around
(and drop the
from the notation for
shortness)
hence
The
is estimated almost exactly the same
way
The rest follows similarly to the proof of the proposition
(
Backward Euler convergence 2
).