Accuracy-Enhancement of Discontinuous Galerkin Methods for PDEs Containing High Order Spatial Derivatives

In this paper, we consider the accuracy-enhancement of discontinuous Galerkin (DG) methods for solving partial differential equations (PDEs) with high order spatial derivatives. It is well known that there are highly oscillatory errors for finite element approximations to PDEs that contain hidden superconvergence points. To exploit this information, a Smoothness-Increasing Accuracy-Conserving (SIAC) filter is used to create a superconvergence filtered solution. This is accomplished by convolving the DG approximation against a B-spline kernel. Previous theoretical results about this technique concentrated on first- and second-order equations. However, for linear higher order equations, Yan and Shu (J Sci Comput 17:27–47, 2002) numerically demonstrated that it is possible to improve the accuracy order to 2 k + 1 for local discontinuous Galerkin (LDG) solutions using the SIAC filter. In this work, we firstly provide the theoretical proof for this observation. Furthermore, we prove the accuracy order of the ultra-weak local discontinuous Galerkin (UWLDG) solutions could be improved to 2 k + 2 - m using the SIAC filter, where m = [ n 2 ] , n is the order of PDEs. Finally, we computationally demonstrate that for nonlinear higher order PDEs, we can also obtain a superconvergence approximation with the accuracy order of 2 k + 1 or 2 k + 2 - m by convolving the LDG solution and the UWLDG solution against the SIAC filter, respectively.