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

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 Smoothness-Increasing 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

One-Sided Position-Dependent Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering Over Uniform and Non-Uniform Meshes

In this paper, we introduce a new position-dependent Smoothness-Increasing Accuracy-Conserving (SIAC) filter that retains the benefits of position dependence while ameliorating some of its shortcomings. As in the previous position-dependent filter, our new filter can be applied near domain boundaries, near a discontinuity in the solution, or at the interface of different mesh sizes; and as before, in general, it numerically enhances the accuracy and increases the smoothness of approximations obtained using the discontinuous Galerkin (dG) method. However, the previously proposed position-dependent one-sided filter had two significant disadvantages: (1) increased computational cost (in terms of function evaluations), brought about by the use of $4k+1$ central B-splines near a boundary (leading to increased kernel support) and (2) increased numerical conditioning issues that necessitated the use of quadruple precision for polynomial degrees of $k\ge 3$ for the reported accuracy benefits to be realizable numerically. Our new filter addresses both of these issues --- maintaining the same support size and with similar function evaluation characteristicsas the symmetric filter in a way that has better numerical conditioning --- making it, unlike its predecessor, amenable for GPU computing. Our new filter was conceived by revisiting the original error analysis for superconvergence of SIAC filters and by examining the role of the B-splines and their weights in the SIAC filtering kernel. We demonstrate, in the uniform mesh case, that our new filter is globally superconvergent for $k=1$ and superconvergent in the interior (e.g., region excluding the boundary) for $k\ge2$. Furthermore, we present the first theoretical proof of superconvergence for postprocessing over smoothly varying meshes, and explain the accuracy-order conserving nature of this new filter when applied to certain non-uniform meshes cases. We provide numerical examples supporting our theoretical results and demonstrating that our new filter, in general, enhances the smoothness and accuracy of the solution. Numerical results are presented for solutions of both linear and nonlinear equation solved on both uniform and non-uniform one- and two-dimensional meshes

Residual estimates for post-processors in elliptic problems

In this work we examine a posteriori error control for post-processed approximations to elliptic boundary value problems. We introduce a class of post-processing operator that `tweaks' a wide variety of existing post-processing techniques to enable efficient and reliable a posteriori bounds to be proven. This ultimately results in optimal error control for all manner of reconstruction operators, including those that superconverge. We showcase our results by applying them to two classes of very popular reconstruction operators, the Smoothness-Increasing Accuracy-Enhancing filter and Superconvergent Patch Recovery. Extensive numerical tests are conducted that confirm our analytic findings.Comment: 25 pages, 17 figure

Hexagonal Smoothness-Increasing Accuracy-Conserving Filtering

Discontinuous Galerkin (DG) methods are a popular class of numerical techniques to solve partial differential equations due to their higher order of accuracy. However, the inter-element discontinuity of a DG solution hinders its utility in various applications, including visualization and feature extraction. This shortcoming can be alleviated by postprocessing of DG solutions to increase the inter-element smoothness. A class of postprocessing techniques proposed to increase the inter-element smoothness is SIAC filtering. In addition to increasing the inter-element continuity, SIAC filtering also raises the convergence rate from order k+1k+1 to order 2k+12k+1 . Since the introduction of SIAC filtering for univariate hyperbolic equations by Cockburn et al. (Math Comput 72(242):577–606, 2003), many generalizations of SIAC filtering have been proposed. Recently, the idea of dimensionality reduction through rotation has been the focus of studies in which a univariate SIAC kernel has been used to postprocess a two-dimensional DG solution (Docampo-Sánchez et al. in Multi-dimensional filtering: reducing the dimension through rotation, 2016. arXiv preprint arXiv:1610.02317). However, the scope of theoretical development of multidimensional SIAC filters has never gone beyond the usage of tensor product multidimensional B-splines or the reduction of the filter dimension. In this paper, we define a new SIAC filter called hexagonal SIAC (HSIAC) that uses a nonseparable class of two-dimensional spline functions called hex splines. In addition to relaxing the separability assumption, the proposed HSIAC filter provides more symmetry to its tensor-product counterpart. We prove that the superconvergence property holds for a specific class of structured triangular meshes using HSIAC filtering and provide numerical results to demonstrate and validate our theoretical results

Smoothness-Increasing Accuracy-Conserving (SIAC) filtering and quasi interpolation: A unified view

Filtering plays a crucial role in postprocessing and analyzing data in scientific and engineering applications. Various application-specific filtering schemes have been proposed based on particular design criteria. In this paper, we focus on establishing the theoretical connection between quasi-interpolation and a class of kernels (based on B-splines) that are specifically designed for the postprocessing of the discontinuous Galerkin (DG) method called Smoothness-Increasing Accuracy-Conserving (SIAC) filtering. SIAC filtering, as the name suggests, aims to increase the smoothness of the DG approximation while conserving the inherent accuracy of the DG solution (superconvergence). Superconvergence properties of SIAC filtering has been studied in the literature. In this paper, we present the theoretical results that establish the connection between SIAC filtering to long-standing concepts in approximation theory such as quasi-interpolation and polynomial reproduction. This connection bridges the gap between the two related disciplines and provides a decisive advancement in designing new filters and mathematical analysis of their properties. In particular, we derive a closed formulation for convolution of SIAC kernels with polynomials. We also compare and contrast cardinal spline functions as an example of filters designed for image processing applications with SIAC filters of the same order, and study their properties

Doctor of Philosophy

dissertationSmoothness-increasing accuracy-conserving (SIAC) filters were introduced as a class of postprocessing techniques to ameliorate the quality of numerical solutions of discontinuous Galerkin (DG) simulations. SIAC filtering works to eliminate the oscillations in the error by introducing smoothness back to the DG field and raises the accuracy in the L2-n o rm up to its natural superconvergent accuracy in the negative-order norm. The increased smoothness in the filtered DG solutions can then be exploited by simulation postprocessing tools such as streamline integrators where the absence of continuity in the data can lead to erroneous visualizations. However, lack of extension of this filtering technique, both theoretically and computationally, to nontrivial mesh structures along with the expensive core operators have been a hindrance towards the application of the SIAC filters to more realistic simulations. In this dissertation, we focus on the numerical solutions of linear hyperbolic equations solved with the discontinuous Galerkin scheme and provide a thorough analysis of SIAC filtering applied to such solution data. In particular, we investigate how the use of different quadrature techniques could mitigate the extensive processing required when filtering over the whole computational field. Moreover, we provide detailed and efficient algorithms that a numerical practitioner requires to know in order to implement this filtering technique effectively. In our first attempt to expand the application scope of this filtering technique, we demonstrate both mathematically and through numerical examples that it is indeed possible to observe SIAC filtering characteristics when applied to numerical solutions obtained over structured triangular meshes. We further provide a thorough investigation of the interplay between mesh geometry and filtering. Building upon these promising results, we present how SIAC filtering could be applied to gain higher accuracy and smoothness when dealing with totally unstructured triangular meshes. Lastly, we provide the extension of our filtering scheme to structured tetrahedral meshes. Guidelines and future work regarding the application of the SIAC filter in the visualization domain are also presented. We further note that throughout this document, the terms postprocessing and filtering will be used interchangeably

Smoothness-increasing accuracy-conserving (SIAC) line filtering: effective rotation for multidimensional fields

Over the past few decades there has been a strong effort towards the development of Smoothness-Increasing Accuracy-Conserving (SIAC) filters for Discontinuous Galerkin (DG) methods, designed to increase the smoothness and improve the convergence rate of the DG solution through this post-processor. The applications of these filters in multidimension have traditionally employed a tensor product kernel, allowing a natural extension of the theory developed for one-dimensional problems. In addition, the tensor product has always been done along the Cartesian axis, resulting in a filter whose support has fixed shape and orientation. This thesis has challenged these assumptions, leading to the investigation of rotated�filters: tensor product filters with variable orientation. Combining this approach with previous experiments on lower-dimension filtering, a new and computationally efficient subfamily for post-processing multidimensional data has been developed: SIAC Line filters. These filters transform the integral of the convolution into a line integral. Hence, the computational advantages are immediate: the simulation times become significantly shorter and the complexity of the algorithm design reduces to a one-dimensional problem. In the thesis, a solid theoretical background for SIAC Line �filters has been established. Theoretical error estimates have been developed, showing how Line filtering preserves the properties of traditional tensor product �filtering, including smoothness recovery and improvement in the convergence rate. Furthermore, different numerical experiments were performed, exhibiting how these filters achieve the same accuracy at significantly lower computational costs. This affords great advantages towards the applications of these filters during flow visualization; one important limiting factor of a tensor product structure is that the filter grows in support as the field dimension increases, becoming computationally expensive. SIAC Line filters have proven effi�ciency in computational performance, thus overcoming the limitations presented by the tensor product filter. The experiments carried out on streamline visualization suggest that these filters are a promising tool in scientific visualisation