Error Analysis of Higher Order Bivariate Lagrange and Triangular Interpolations in Electromagnetics

Abstract

The interpolation errors of the higher order bivariate Lagrange polynomial interpolation based on the rectangular, right and equilateral triangular interpolations are measured by using the maximum and root-mean-square (RMS) errors. The error distributions of above three kinds of interpolations are analyzed to find the regions having the smallest interpolation error. Both analytical and numerical results show that the right triangular interpolation is the most efficient interpolation method. Although both the maximum and RMS errors of the right triangular interpolation are larger than that of the rectangular interpolation, the number of data points used in the triangular interpolation is up to 50% less than that used in the rectangular one. On the other hand, the equilateral triangular interpolation using the regions inside a big triangle as interpolation area is proved to be the most accurate interpolation method. The interpolation errors of the equilateral triangular are almost half of that of the rectangular interpolation. In addition, the right triangular, equilateral triangular and rectangular interpolations for the third and fourth orders are applied to accelerate the calculation of doubly periodic Green's function (PGF). Index Terms-Doubly periodic Green's function (PGF); Lagrange polynomial interpolation; maximum error; plane waves; root-mean-square (RMS) error; triangular interpolations. I. INTRODUCTION NTERPOLATION has been widely used in many areas of computational electromagnetics (CEM) to save the CPU time, e.g., the moment method, the finite element method (FEM), and so on [1-5]. Accurate and efficient evaluation of free-space periodic Green's functions (PGF) takes an important role in CEM [6, 7]. The PGF with two-dimensional (2-D) lattices is in the form of double infinite spectral and spatial summations, which is usually slowly convergent [8, 9]. For the doubly PGF,

Dispersion Characteristics and Applications of Higher Order Isosceles Triangular Meshes in the Finite Element Method

“…Luo et al found the advantage of triangular interpolation method by selecting the interpolation region with less error. Compared with the traditional interpolation based on rectangular meshes, the right triangular interpolation method is more effective, and the equilateral triangular interpolation method is more accurate [13], [14]. Recently, Liu and Luo et al proposed an improved method based on non-uniform isosceles triangular interpolation to accelerate the solution of multilevel fast multipole algorithm (MLFMA) [15].…”

Harmonic Vector Error Analysis Based on Lagrange Interpolation

“…The Lagrange error must be minimized. This algorithm is used in the interval, and the interpolation accuracy increases within a certain range by increasing the number of interpolation points [8]. Reference [9] introduced an effective digital calibration algorithm based on Lagrange interpolation for the phase shift factor of electronic meters, which reduces the phase shift of static meters.…”