Efficient Triangular Interpolation Methods: Error Analysis and Applications

Abstract

The interpolation errors of bivariate Lagrange polynomial and triangular interpolations are studied for the plane waves. The maximum and root-mean-square (rms) errors on the right triangular, equilateral triangular, and rectangular (bivariate Lagrange polynomial) interpolations are analyzed. It is found that the maximum and rms errors are directly proportional to the ( p + 1)th power of kh for both 1-D and (2-D, bivariate) interpolations, where k is the wavenumber and h is the mesh size. The interpolation regions for the right triangular, equilateral triangular, and rectangular interpolations are selected based on the regions with smallest errors. The triangular and rectangular interpolations are applied to evaluate the 2-D singly periodic Green's function. The numerical results show that the equilateral triangular interpolation is the most accurate interpolation method, whereas the right triangular interpolation is the most efficient interpolation method.