Convergence rate estimation of poly-Sinc-based discontinuous Galerkin methods

Abstract

The error of approximation of a function using Poly-Sinc approximation was shown to have a convergence rate of exponential order over a global partition. However, it was assumed that the function values at the interpolation points were known. In this paper, we extend our work on poly-Sinc-based discontinuous Galerkin approximation, in which the function values at the interpolation points are replaced by unknown coefficients and solved by the discontinuous Galerkin method. To deal with functions having a singularity at an endpoint, we use Sinc partitions in our discontinuous Galerkin approximation. For such functions, we show that, by using a weighted L2 norm over a Sinc partition in which the weight function is the reciprocal of the asymptotic density of Sinc points in that partition and computing the ℓ2 norm over all Sinc partitions constituting the global partition except the partition containing the singularity, the error of approximation between the exact solution of the ordinary differential equation and its poly-Sinc-based discontinuous Galerkin approximation has a convergence rate of exponential order similar to the one over the global partition. The numerical results are in agreement with our theoretical derivations of the approximation error. We compare our poly-Sinc-based discontinuous Galerkin method with the poly-Sinc collocation method. Our method shows better performance in the L2 norm.