e apply technique of the section (
Finite
elements
) and results of the section
(
Calculation
of approximation spaces in one dimension
) to the following problem.
According to the section
(
Representation
of solution for elliptic PDE using stochastic process
), the function
is also a solution of the following problem.
Problem
(Example strong elliptic problem)
Find
s.t.
According to the section (
Elliptic PDE
),
solution of the problem (
Example
strong elliptic problem
) may be calculated by solving the weak problem.
Problem
(Example weak elliptic problem) Find
s.t.
Following recipes of the section
(
Finite
elements for Poisson equation with Dirichlet boundary conditions
) we seek
the solution of the
form
where
is a selection from a wavelet basis on
.
We take
for
from the same selection and arrive to the following approximate
problem.
Selection process for the basis
is already introduced in the section
(
Decomposition
of payoff function in one dimension
).
Problem
(Example approximate problem) Find
s.t.
The procedure is performed in the script
OTSProject/python/wavelet2/poisson.py. We
chose
For such choice we verify the answer
directly:
We apply the operation
.
Thus
where the constants
,
are chosen to satisfy
Thus
or
We add and subtract the
equation,
Using 45 basis functions
we achieve the
precision
