Computing eigenmodes of elliptic operators using increasingly flat radial basis functions

Solving multi-dimensional eigenmodes problem for elliptic operator using radial basis functions (RBFs) was proposed by Platte and Driscoll (2004) [14]. They convert the eigenmodes problem to an eigenpairs problem of a finite dimensional matrix. We formulate an approach based on using finite order interpolating polynomials as eigenfunctions for eigenmodes problem. We prove that, under some simple conditions on the RBFs, two approaches converge when increasingly flat BRFs are being used. These results are supported by numerical examples.