

This is the current state of my research and intentions as of the date selected. 
Finite Element to Finite Difference Interpolation
Although the initial data is most easily evolved using a finite element method, specially adapted to the distorted apparent horizon, it is simpler to use a finite difference method for the hyperbolic evolution equations. This means that the irregular finite element mesh should be interpolated onto a regular rectangular grid where the numerical derivatives will be well defined. It is necessary that the interpolated values still satisfy the constraint equations to the same order of accuracy.
The finite element solution is a well defined interpolant of the solution, and naively one would believe that this solution could be numerically differentiated in such a way that it still solves the constraint equations. On closer inspection, the irregular mesh wreaks havoc on the postprocessing differentiation and leads to order one quantities in the residual.
This is described in more detail here. The solution is simply to use a higher order interpolation operator, say a bicubic interpolation calculated with the ten nearest neighbors to the point desired to interpolate. 
My work is funded by NSERC/CRSNG. 