Superconvergent error estimates for position-dependent smoothness-increasing accuracy-conserving (SIAC) post-processing of discontinuous Galerkin solutions

Abstract:

Superconvergence of discontinuous Galerkin methods is an area of increasing interest due to the ease with which higher order information can be extracted from the approximation. Cockburn, Luskin, Shu, and Süli showed that by applying a B-spline filter to the approximation at the final time, the order of accuracy can be improved from $\mathcal {O}(h^{k+1})$ to $\mathcal {O}(h^{2k + 1})$ in the $\mathcal {L}^{2}$-norm for linear hyperbolic equations with periodic boundary conditions (where $k$ is the polynomial degree and $h$ is the mesh element diameter) [Math. Comp. (2003)]. The applicability of this filter for linear hyperbolic problems with non-periodic boundary conditions was computationally extended and renamed a position-dependent smoothness-increasing accuracy-conserving (SIAC) filter by van Slingerland, Ryan, Vuik [SISC (2011)]. However, error estimates in the $\mathcal {L}^2$-norm for this new position-dependent SIAC filter were never given. Furthermore, error estimates in the $\mathcal {L}^\infty$-norm have not been established for the original kernel nor the position-dependent kernel. In this paper, for the first time we establish that it is possible to obtain $\mathcal {O}(h^{\min \{2k+1,2k + 2-\frac {d}{2}\}})$ accuracy in the $\mathcal {L}^{\infty }$-norm for the position-dependent SIAC filter, where $d$ is the dimension. Furthermore, we extend the error estimates given by Cockburn et al. so that they are applicable to the entire domain when implementing the position-dependent SIAC filter. We also computationally demonstrate the applicability of this filter for visualization of streamlines.