The Chebyshev spectral element approximation with exact quadratures

•The paper addressed and assessed some of the issues concerning the numerical accuracy on the Chebyshev spectral element method.•A new Chebyshev spectral element method is developed by using exact quadratures in computing spectral elements.•The method is validated with the Stokes and the Cauchy–Riemann problems.•Numerical results show that an enhancement of the approximation convergence rate is attained.•Numerical accuracy is much better than that from other spectral element methods.

A new Chebyshev spectral element method has been developed in this paper, in which exact quadratures are used to overcome a shortfall of the Gauss–Chebyshev quadrature in variational spectral element projections. The method is validated with the Stokes and the Cauchy–Riemann problems. It is shown that an enhancement of the approximation convergence rate is attained, and numerical accuracy is much better than that from the Gauss–Lobatto–Legendre spectral element method.