The discontinuous Galerkin (DG) method has recently become one of the most widely researched and utilized discretization methodologies for solving problems in science and engineering. It is fundamentally based upon the mathematical framework of variational methods, and provides hope that computationally fast, efficient and robust methods can be constructed for solving real-world problems. By not requiring that the solution to be continuous across element boundaries, DG provides a flexibility that can be exploited both for geometric and solution adaptivity and for parallelization. This freedom comes at a cost. Lack of smoothness across elements can hamper simulation post-processing like feature extraction and visualization. However, these limitations can be overcome by taking advantage of an additional property of DG - that of superconvergence. Superconvergence can aid in addressing the lack of continuity through the development of Smoothness-Increasing Accuracy-Conserving (SIAC) filters. These filters respect the mathematical properties of the data while providing levels of smoothness so that commonly used visualization tools can be used appropriately, accurately, and efficiently. In this talk, the importance of superconvergence in applications such as visualization will be discussed as well as overcoming the mathematical barriers in making superconvergence useful for applications.
