A novel Chebyshev collocation method for elliptic -type differential equations with degenerate coefficient

Abstract

A novel collocation scheme is presented for elliptic-type differential equations with degenerate coefficients and homogeneous Dirichlet boundary conditions. The use of weighted orthogonal Chebyshev polynomials for the basis functions leads to stiffness matrices with sparse structure, enabling efficient direct calculations. By an orthogonal projection, rigorous analyses are devoted to deriving a-priori error estimates of spectral accuracy in two norms. Furthermore, ample numerical experiments are conducted and compared with error data, convergence rates, condition numbers and $N$-$\log$ curves to confirm the theoretical analyses results. Our proposed method achieves spectral accuracy and handles boundary singularities efficiently, as demonstrated by theoretical analyses and numerical experiments.