his section is of academic significance. To derive a preconditioner
from asymptotic decomposition is a very natural attempt. The author feels
compelled to document it even so the final result could be guessed. The reader
may skip this section without consequence.
According to the section
(
Reduction
to system of linear algebraic equations for q=1
), we need to solve the
equation
where
Thus, we transform
into the set of coefficients
such
that
is an approximation to the solution
of the
problem
taken at
:
In the section
(
Asymptotic
expansion for Black equation
) we calculated an approximate solution to the
problem
in the
form
Thus we have an explicit transformation
where
is close to
.
We use such transformation to construct a preconditioner
for the
step
of the chain
.
It is essential that
are constructed as piecewise quadratic functions in
,
see the sections
(
Calculation of boundary
scaling functions
),
(
Calculation
of approximation spaces in one dimension II
). Thus, the first derivative
is continuous and piecewise linear function and the second derivative is a
piecewise constant function. We do not reach a situation when we would have to
keep track of delta functions. We can apply facilities of PiecewisePoly
library (see the section
(
Manipulation
of localized piecewise polynomial functions
)) without additional
modification. It also helps to do integration by part to even out
differentiation.
For every
consider
the
transformation
where the
might be outside of
span
.
The sum should be understood as a projection. We put together the matrix
Hence, the semisolution
would
act
Thus
The
is represented by
where
is the
Gram
matrix
and
is a column
.
Thus we arrive to the
chain
Therefore, we
set
The projection in the above summary is calculated as follows.
We introduce the convenience notation
:
Then the coefficients
come from the
relationships
for the
scalar product
.
Thus
or
The expression for the preconditioner
is
thus
It remains to improve the calculation of
:
where the functions
are piecewise polynomials in
vanishing at the ends of integration interval. We integrate by
parts:
and arrive to the same expression we already use when constructing the matrix
.
Thus
while the matrix
is given
by
Hence the preconditioned matrix
is
The matrix
is
symmetric.
