I have sets of multipoint field data, each point data set relates to a single cell of an unstructured mesh. The goal is to interpolate the data to the cell centre, directly or indirectly, in the most accurate way.
If I use Inverse Distance Weighted interpolation, in the case when the distance between the source and the target (cell centre) is very small, I may end up with a floating point exception.
For this kind of interpolation on a structured mesh, a volume weighted interpolation is used. This does not translate directly to an arbitrarily shaped mesh cell.
Introducing a tolerance for a IDW interpolation to circumvent the SIGFPE makes sense only if I do not introduce any tests that could render the interpolation inefficient. Is adding a sufficiently small $\delta$ to the denominator for every weight a possible option with the IDW interpolation? What interpolation methods suitable for this problem do you know?
Additional info:
For the interpolation from the mesh to the points, I am using an interpolation based on the barcycentric coordinates. Each polyhedral cell of the mesh is decomposed into tetrahedra. Cell centred field is interpolated to the cell points using IDW interpolation. A search is conducted for each point to find the tetrahedron within which it lies, and the values are interpolaed using the barycentric interpolation.
For the interpolation from the points to the mesh, this is not possible. The cell centred values are unknown. There is no way to assemble a tetrahedral composition that would enforce $\sum_p W_{PC} = 1$, where $W_{PC}$ is the weight related to a point P and a cell centre C. This comes from the fact that the point configuration is arbitrary. So, I am currently using IDW for this, making sure that I don't get a floating point exeption. Are there any better suited interpolation methods for this problem?