uppose
is
a closed domain in
.
Let
be a lattice with step
covering
.
We introduce functions
with domain in
and range in
.
Finite difference
operator
is operator acting on such functions. The regular notation is
.
We write
and denote
the restriction of a regular function
on
Sometimes we also use notation
for a function defined on
.
For example,
The finite difference operator
approximates
on the function
with order
if there exist constants
such that for any
Similarly, one defines approximation of a function
by a function
by means of the inequality
For example, the defined above finite difference operator
approximates
with
order 2 on any smooth function.
The finite difference problem
("scheme")
is
stable
if there exist constants
such that for any
the solution
satisfies
The following theorem (called "Lax convergence theorem")
states that if the
scheme approximates and is stable then it converges to the right solution.
