Discontinuous Galerkin methods using poly-sinc approximation

Abstract

In this paper, we develop a discontinuous Galerkin method using poly-Sinc approximation, in which non-equidistant points are generated by conformal mappings, such as Sinc points. Sinc points are characterized by their dense distribution near the endpoints of an interval, which render them effective in dealing with jumps introduced by the partitioning process of the interval in a discontinuous Galerkin scheme. Sinc methods are characterized by their fast convergence rate of exponential order and effective handling of singularities. We use equidistant, Chebyshev, and Sinc partitions to handle the singularities. We demonstrate that, using poly-Sinc-based discontinuous Galerkin approximation, Sinc partitions outperforms those with equidistant and Chebyshev partitions as we near the singularity. We also demonstrate that for Sinc partitions, poly-Sinc-based discontinuous Galerkin approximation outperforms those with Chebyshev points. We used the stability condition of the discontinuous Galerkin method to demonstrate the stability of our approach. It was shown that the approximation error between a smooth function and its poly-Sinc approximation over the global partition has a convergence rate of exponential order. For a function with a singularity at an endpoint, we use the weighted L2 norm over the Sinc partitions. Using the weighted L2 norm, we show that, by computing the ℓ2 norm over all Sinc partitions constituting the global partition except the last partition, the approximation error between the exact solution of the ordinary differential equation and its poly-Sinc-based discontinuous Galerkin approximation has a convergence rate of exponential order.