Convergence of Wachspress coordinates: from polygons to curved domains

Abstract

Given a smooth, strictly convex planar domain, we investigate point-wise convergence of the sequence of Wachspress coordinates defined over finer and finer inscribed polygonal approximations of the domain. Based on a relation between the discrete Wachspress case and the limit smooth case given by the Wachspress kernel defined by Warren et al., we show that the corresponding sequences of Wachspress interpolants and mappings converge as O(h 2 ) for a sampling step size h of the boundary curve of the domain as h → 0. Several examples are shown to numerically validate the results and to visualise the behaviour of discrete interpolants and mappings as they converge to their smooth counterparts. Empirically, the same convergence order is observed also for mean value coordinates. Moreover, our numerical tests suggest that the convergence of interpolants and mappings is uniform both in the Wachspress and mean value cases.

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

“…It is easy to see that (19) reproduces exactly constant and linear functions. In contrary to solutions computed with the finite element method, (19) can be evaluated pointwise without the need to mesh the computational domain.…”

“…While the approximations delivered by (19) are good in these examples, it may not always produce a sufficiently accurate approximation. For example, the approximation error for the ring domain (see Fig.…”

“…In contrary to solutions computed with the finite element method, (19) can be evaluated pointwise without the need to mesh the computational domain. Additionally, for complicated domains or functions were (19) does not deliver a sufficiently accurate solution, it can be used as the initial solution to an iterative scheme (such as Newton-Raphson) typically required for solving the nonlinear p-Laplace equation.…”

“…We evaluate numerically our interpolation scheme (19) and compare it to the fundamental solution f (x) given by (20) for different values of p on a concave shape shown in Fig. 12.…”

Convergence of barycentric coordinates to barycentric kernels

“…Considering denser and denser polygons inscribed into a smooth domain leads to a (convergent) series of barycentric coordinates. This limit process was recently investigated in the case of Wachspress coordinates in Kosinka and Bartoň (2015). It was shown that in the limit, as the number of vertices of the polygon approaches infinity, one obtains the Wachspress kernel Warren et al (2007) introduced by Warren et al, and that the convergence rate is quadratic.…”