A new collocation method using near-minimal Chebyshev quadrature nodes on a square

Abstract

A new collocation method using near-minimal Chebyshev quadrature nodes is proposed on the square [−1,1]2. For a (total) degree of precision 2n−1, the number of nodes of this near-minimal quadrature rule amounts only to n(n+1)2+⌊n2⌋+1, which is one more than the Möller's lower bound, n(n+1)2+⌊n2⌋, i.e., the minimal number of nodes in a quadrature rule of degree 2n−1 in two dimensions. Firstly, a new Chebyshev interpolation based on the near-minimal Chebyshev quadrature rule is constructed. An optimal error estimate on the new interpolation is obtained, and fast algorithms for the corresponding discrete Chebyshev transformation (DCT) between the function values and the discrete Chebyshev coefficients are then devised. Next, spectral differentiation schemes are developed both in the physical space and in the frequency space. Finally, a new Chebyshev collocation method, which uses nearly half nodes of the tensorial Chebyshev collocation method, is proposed to solve second order partial differential equations on the square. Numerical experiments illustrate that our new Chebyshev collocation method also possesses an exponential order of convergence for smooth problems. In comparison to the tensorial collocation method, it can offer better accuracy for most problems with the same degrees of freedom.