There are two problems with this kind of procedure. First, the Yamabe equation degenerates on the boundary. This is a difficult problem if one tries to prove existence of solutions of this equation, because the loss of ellipticity on the boundary means that one cannot simply appeal to known theorems. However, numerically this has not been a serious obstacle. The Yamabe equation is solved in a more or less standard way by using a Richardson iteration scheme to reduce the solution of the non-linear equation to a series of inversions of a linear operator.
The degeneracy of the equation on the boundary forces the solution and its derivative by regularity to have certain well defined boundary values. This and the global nature of the elliptic equations suggests that we use pseudo-spectral (or collocation) methods. Such methods are well suited for problems for which it is known beforehand that the solution will be sufficiently regular. Then, the solution can be expanded into a series of certain basis functions that are globally analytic. Therefore, the regularity conditions on the boundary are already built into the method. We refer readers interested in these methods to a recent review paper by Bonazzola et al. [25].
The more difficult problem in the determination
of the initial data is the fact that the curvature components are
obtained by successive division by . Since
vanishes on the boundary one has to compute a value
which is of the form
. Although the theorem tells
us that this is well defined (provided that some boundary
conditions are satisfied), it still poses a numerical problem,
because a straightforward implementation of l’Hôpital’s rule loses
accuracy due to the truncation error in the calculation of the
numerator.
There exist two methods to overcome this problem.
The first one [47, 51] is again based on
pseudo-spectral methods. Essentially, in this approach the
multiplication with the function
is expressed as a
linear operation between the expansion coefficients of
and the other factor. The pseudo-inversion of this
linear operator (in the sense of finding its Moore-Penrose
inverse [123]) then corresponds to
division by
.
A different method for computing the quotient has
been devised by Hübner [95]. Here, the problem
of dividing by
has been reformulated into a problem
for solving a second-order PDE for the quotient. This PDE is
similar to the Yamabe equation in the sense that it also
degenerates on the boundary where
vanishes, but it
is linear.
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Another consequence of the singularity
of the Yamabe equation on is the following. It is
usually desirable to extend the solution of the Yamabe equation
beyond
in such a way that it remains a solution of the
equation. The reason is mainly that
is usually not
aligned with the boundary of the computational domain so that it is
impossible to restrict the calculation to the physical domain
inside
. Furthermore, the numerical stencils used in the
discretisation of the evolution equations usually extend beyond
into the unphysical part of the computational
domain, thus picking up information from the outside. Therefore, if
the constraints are not satisfied outside
then it might be possible that constraint violating
modes are excited. While the solution of the Yamabe equation inside
is uniquely determined, this
is not so on the outside, where the
solution is influenced by the choice of boundary conditions on the
boundary of the computational grid. This means that the solution
cannot be smooth across
unless the boundary
conditions are chosen exactly right. But in general it is
impossible to know the correct boundary values beforehand. It is
guaranteed that the solution will be at least
across
, but the third derivative may be
discontinuous.
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This situation does in fact occur as
illustrated in Figure 12, which shows in the
top picture a solution
of the Yamabe equation in an
axi-symmetric space-time. The
-axis (
) is the axis of rotation, and the
-axis (
) corresponds to the
equatorial plane. The solution is assumed to be equatorially
symmetric, so that it is obtained by specifying
on the boundary
and
while using the symmetry conditions on the other
parts of the computational grid. On the bottom are shown the third
differences of
in the
-direction. It is
clearly visible that
is located at the quarter circle
. In Figure 13
we illustrate the
“shielding” property of
for solutions of the Yamabe
equation. Here is shown the difference of two solutions obtained
with different boundary data on the outside. The solutions agree to
a very high accuracy inside the physical domain. Some of these
issues and possible ways to overcome them are discussed in further
detail in [51].