On Multivariate Chebyshev Polynomials and Spectral Approximations on Triangles

In this paper we describe the use of multivariate Chebyshev polynomials in computing spectral derivations and Clenshaw–Curtis type quadratures. The multivariate Chebyshev polynomials give a spectrally accurate approximation of smooth multivariate functions. In particular we investigate polynomials derived from the A2 root system. We provide analytic formulas for the gradient and integral of A2 bivariate Chebyshev polynomials. This yields triangular based Clenshaw–Curtis quadrature and spectral derivation algorithms with O(N logN) computational complexity. Through linear and nonlinear mappings, these methods can be applied to arbitrary triangles and non-linearly transformed triangles. A MATLAB toolbox and a C++ library have also been developed for these methods.