On modified Gordon-Wixom interpolation schemes and their applications to nonlinear and exterior domain problems

Abstract

We introduce and study extensions and modifications of the GordonWixom transfinite barycentric interpolation scheme (Gordon and Wixom, SIAM J. Numer. Anal. 11(5), 1974). We demonstrate that the modified GordonWixom scheme proposed in Belyaev and Fayolle (Comput. Graph. 51, [74][75][76][77][78][79][80] 2015) reproduces harmonic quadratic polynomials in convex domains. We adapt the scheme for dealing with the exterior of a bounded domain and for the exterior of a disk, where we demonstrate that our interpolation formula reproduces harmonic functions. Finally, we show how to adapt the Gordon-Wixom approach for approximating p-harmonic functions and to derive computationally efficient approximations of the solutions to boundary value problems involving the p-Laplacian.