Boundary correction for piecewise linear interpolation defined over data-dependent triangulations

Given a set of data points in R² and corresponding data values it is clear that the quality of a Piecewise Linear Interpolating Surface (PLIS) over triangles depends on the specific triangulation of the data points. While conventional triangulation methods depend only on the distribution of the data points in R², the authors (1990) suggested to construct data-dependent triangulations which depend on the data values as well. Numerical examples indicate that this method improves the PLIS only away from the boundary of the triangulation. In this paper we present and test a numerical scheme for boundary correction to be used as a complementary step to the creation of data-dependent triangulations. This scheme adds more points on the boundary edges of "bad" triangles and estimates corresponding function values. Numerical tests show the success of the scheme in improving the PLIS also near the boundary of the triangulation.