Exploiting Superconvergence Through Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering

Abstract

There has been much work in the area of superconvergent error analysis for finite element and discontinuous Galerkin (DG) methods. The property of superconvergence leads to the question of how to exploit this information in a useful manner, mainly through superconvergence extraction. There are many methods used for superconvergence extraction such as projection, interpolation, patch recovery and B-spline convolution filters. This last method falls under the class of SmoothnessIncreasing Accuracy-Conserving (SIAC) filters. It has the advantage of improving both smoothness and accuracy of the approximation. Specifically, for linear hyperbolic equations it can improve the order of accuracy of a DG approximation from k + 1 to 2k + 1, where k is the highest degree polynomial used in the approximation, and can increase the smoothness to k − 1. In this article, we discuss the importance of overcoming the mathematical barriers in making superconvergence extraction techniques useful for applications, specifically focusing on SIAC filtering.

Residual Estimates for Post-processors in Elliptic Problems

“…In order to show the versatility of our results, we consider two families of reconstruction operators. Namely, the Smoothness-Increasing Accuracy-Conserving (SIAC) post-processing [5,30,32] as well as patch reconstruction via the Zienkiewicz and Zhu [37,39] Superconvergent Patch Recovery (SPR) technique. Below we outline the procedure for performing these reconstructions as well as error estimates for the ideal case.…”

Multi-Dimensional Filtering: Reducing the Dimension Through Rotation

“…Before introducing the filter rotation, we briefly review the original post-processor from which it derives. For a much more detailed description on the properties and implementation of SIAC filters, we refer the reader to [13,14,21,16]. The post-processor is a continuous convolution:…”

“…We do this using a new approach to Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering, which we call SIAC Line filtering. SIAC Filters [21] are a post-processing technique designed to accelerate the convergence rate and increase the smoothness of DG solutions. Traditional applications of SIAC filters require a tensor product construction.…”

“…

Over the past few decades there has been a strong effort towards the development of Smoothness-Increasing Accuracy-Conserving (SIAC) filters [21] for Discontinuous Galerkin (DG) methods, designed to increase the smoothness and improve the convergence rate of the DG solution through this post-processor. These advantages can be exploited during flow visualization, for example by applying the SIAC filter to the DG data before streamline computations [Steffan et al, IEEE-TVCG 14(3): 680-692].

…”