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.

中文翻译：

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高效三角插值法：误差分析及应用

研究了平面波的二元拉格朗日多项式和三角插值的插值误差。分析了直角三角形、等边三角形和矩形（二元拉格朗日多项式）插值的最大和均方根 (rms) 误差。发现对于一维和（二维，双变量）插值，最大和均方根误差与 kh 的 (p + 1) 次方成正比，其中 k 是波数，h 是网格大小。直角三角形、等边三角形和矩形插值的插值区域是根据误差最小的区域选择的。三角形和矩形插值用于计算二维单周期格林函数。