Smoothness-Increasing Accuracy-Conserving Filters for Discontinuous Galerkin Methods: Challenging the Assumptions of Symmetry and Uniformity

In this dissertation, we focus on exploiting superconvergence for discontinuous Galerkin methods and constructing a superconvergence extraction technique, in particular, Smoothness-Increasing Accuracy-Conserving (SIAC) filtering. The SIAC filtering technique is based on the superconvergence property of discontinuous Galerkin methods and aims to achieve a solution with higher accuracy order, reduced errors and improved smoothness. The main contributions described in this dissertation are: 1) an efficient one-sided SIAC filter for both uniform and nonuniform meshes; 2) one-sided derivative SIAC filters for nonuniform meshes; 3) the theoretical and computational foundation for using SIAC filters for nonuniform meshes; and 4) the application of SIAC filters for streamline integration. One-sided SIAC filtering is a technique that enhances the accuracy and smoothness of the DG solution near boundary regions. Previously introduced one-sided filters are not directly useful for most applications since they are limited to uniform meshes, linear equations, and the use of multi-precision packages in the computation. Also, the theoretical proofs relied on a periodic boundary assumption. We aim to overcome these deficiencies and develop a new fast one-sided filter for both uniform and nonuniform meshes. By studying B-splines and the negative order norm analysis, we generalized the structure of SIAC filters from a combination of central B-splines to using more general B-splines. Then, a "boundary shape" B-spline (using multiple knots at the boundary) was used to construct a new one-sided filter. We also presented the first theoretical proof of convergence for SIAC filtering over nonuniform meshes (smoothly-varying meshes). One purpose of SIAC filtering is to improve the smoothness of DG solutions. Because of the increased smoothness, we can obtain a better approximation for the derivatives of DG solutions. Derivative filtering over the interior region of uniform meshes was previously studied. However, nonuniform meshes and boundary regions remain a significant challenge. We extended the one-sided filter to a one-sided derivative filter. To deal with nonuniform meshes, we investigated the negative order norm over arbitrary meshes and proposed to scale the one-sided derivative filter with scaling hµ. For arbitrary nonuniform rectangular meshes, we proved that the one-sided derivative filter can enhance the order of convergence for the ?th derivative of the DG solution from k + 1 - ? to µ(2k + 2), where µ ? 2/3. The most challenging part of this project is recovering the superconvergence of the DG solution over nonuniform meshes through SIAC filtering. Typically, most theoretical proofs for SIAC filters are limited to uniform meshes (or translation invariant meshes). The only theoretical investigations for nonuniform meshes were included in our one-sided and derivative filtering studies. Although our earlier research for nonuniform meshes provides good engineering accuracy, we want to do better mathematically. This is not an easy task since unstructured meshes give DG solutions irregular performance under the negative order norm. In our work, we introduced a parameter to measure the unstructuredness of a given nonuniform mesh. Then, by adjusting the scaling of the SIAC filter based on this unstructuredness parameter, we can obtain the optimal filtered approximation (best accuracy) over a given nonuniform mesh. SIAC filtering for streamline integration is an attempt to use SIAC filters in a realistic engineering application. By using the one-sided filter and one-sided derivative filter, we designed an efficient algorithm: filtering the velocity field along the streamline and then use a backward differentiation formula for integration. Compared to the traditional method of filtering the entire field (multi-dimensional algorithm), the computational cost drops dramatically since its complexity corresponds to a one-dimensional algorithm. We finally note that most of the work presented originates from published and submitted papers for the past four years of this PhD research