Reviving the method of particular solutions

Fox, Henrici, and Moler made famous a "method of particular solutions" for computing eigenvalues and eigenmodes of the Laplacian in planar regions such as polygons. We explain why their formulation of this method breaks down when applied to regions that are insufficiently simple and propose a modification that avoids these difficulties. The crucial changes are to introduce points in the interior of the region as well as on the boundary and to minimize a subspace angle rather than just a singular value or a determinant. Similar methods may be used to improve other "mesh-free" algorithms for a variety of computational problems.