chebyshev nodes on a 2D grid

Question

I want to use chebyshev nodes for interpolation using lagrange formula. My grid is two dimensional and i do not know how to determine the nodes of chebyshev in a 2D grid point?

Answer 1

Just use the product of one-dimensional Chebyshev grids, that is, the set of all points $(x_i,y_j)$ where $x_i$ and $y_j$ run over 1D Chebyshev nodes independently. For example, this library uses product Chebyshev grid for polynomial interpolation.

Answer 2

2D Chebyshev points are $\displaystyle{ \left( \cos \frac{n\pi}{N} \,,\, \cos \frac{m\pi}{M} \right) }$ where $n = 0, \dotsm, N$ and $m = 0, \dotsm, M$

They can be generated in Python:

import numpy as np
import matplotlib.pyplot as plt
N = 30
M = 15
x = np.cos (np.pi * np.arange(N + 1) / N)
y = np.cos (np.pi * np.arange(M + 1) / M)
xx, yy = np.meshgrid(x, y)
plt.plot(xx, yy, "ko", markersize = 3)
plt.show()