A short note on the error estimates of Yuan-Shu discontinuous Galerkin method based on non-polynomial approximation spaces

In the article [Yuan and Shu (2006) [17]], Yuan and Shu have developed discontinuous Galerkin (DG) methods based on non-polynomial approximation spaces for solving time dependent problems. The authors established L² error estimates of the proposed methods and also presented a criterion for the choice of basis functions of the non-polynomial spaces to have the same approximation rates as those of polynomial finite element spaces of the same dimension. However, the verification that some of approximation spaces do satisfy the criterion has been performed only to limited types of approximation spaces. In this regards, the aim of this short note is to fill the gap. We prove that the criterion of Yuan and Shu can be satisfied by a wide class of basis functions, including the well-known basis functions such as trigonometric and exponential polynomials.