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 to 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 which 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. [23].

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 0/0. 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 looses accuracy. There have been two methods to overcome this problem. The first one [40] 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 [106]) then corresponds to division by . Although this methods seems to work well, it is not generally applicable because it depends heavily on the geometric setup.

A more general method has been devised by Hübner [82]. 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. Both methods have been applied in 2D test cases.

Conformal Infinity
Jörg Frauendiener
http://www.livingreviews.org/lrr-2000-4
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