I'm working on a finite volume advection scheme for unstructured meshes which uses a multidimensional polynomial weighted least squares fit for interpolating from cell centres onto faces.
In 2D, the advection scheme typically uses a 4x3 stencil of cell values in the domain interior to fit a polynomial that is cubic in x and quadratic in y. The weights resulting from the SVD are expensive to compute but they can be precomputed during initialisation.
There are two cases where which wish to omit some higher order terms: 1. Near boundaries (which may be irregular in shape), the stencil is necessarily smaller 2. If some cell values are very closely spaced on certain meshes, we do not have enough information to sensibly fit high order terms
I'm looking for a technique for dealing with these cases. Can the SVD help me determine which terms to omit? Or are there other techniques I could consider?