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.

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