On transfinite Gordon-Wixom interpolation schemes and their extensions

Among various barycentric coordinates and their extensions, the linear and cubic (Hermite) Gordon-Wixom transfinite interpolation schemes deliver the most accurate approximations of the harmonic and biharmonic functions, respectively. However interpolation properties of the original Gordon-Wixom interpolations are studied for convex domains only and, therefore, their current practical importance is limited. In this paper, we propose simple modifications of the Gordon-Wixom interpolation schemes, study their properties, and show how they can be used for approximating solutions to the Poisson and inhomogeneous biharmonic equations. Our modified Gordon-Wixom interpolations are easily extended to non-convex domains and, according to our experiments, deliver more accurate approximations of the harmonic and biharmonic functions compared with the original Gordon-Wixom schemes. We also demonstrate how our approach can be used for approximating the distance function.