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.@en