Accuracy enhancement of discontinuous Galerkin solutions to hyperbolic PDE's
The cornerstone of my research program is post-processing for discontinuous Galerkin (DG) methods to improve the order of accuracy of the numerical solution. Post-processing for DG is a major area of research, aimed at extracting existing higher order information from the DG solution. The discontinuous Galerkin method utilizes piecewise polynomials of degree k for the approximation space and produces highly oscillatory errors of order k+1. However, the analysis of these methods shows that in the negative order Sobolev norm the error is order 2k+1. This suggests that hidden in the DG solution is higher order information that is not observable due to the oscillatory nature of the errors. Our goal is to extract this higher order information so that it can be computed in a more convenient norm, such as L2. This is done by convolving the approximation against a kernel consisting of B-splines of order k+1. This convolution kernel depends on the DG solution over the entire element, as well as neighboring elements and produces solutions with a higher degree of regularity and smaller errors. I have developed convolution kernels for unstructured meshes, one-sided kernels for use with discontinuities, and kernels that allow extraction of accurate derivative information. Using this convolution kernel restores levels of continuity back into the solution and therefore a natural extension of this technique is visualization filtering.
Filtering for visualization
Although discontinuous and continuous Galerkin methods have advantages mathematically and computationally, they suffer from one “feature” which can in turn become a disadvantage – they do not require high levels of smoothness at the element boundaries. Lack of smoothness across elements can hamper simulation post-processing like feature extraction and visualization. Many commonly used visualization techniques explicitly (or tacitly) assume that the field upon which they are acting is smooth. Applying such techniques under the non-ideal cases of non-smooth solutions can in the best case result only in a loss of convergence rate (or accuracy) and in the worst case can lead to erroneous visualization. Accuracy enhancement techniques such as those of Cockburn, Luskin, Shu, and Suli for the discontinuous Galerkin methods have sought to overcome smoothness issues by using negative order norm estimates to enhance the quality of the solution in terms of both smoothness and accuracy. This research attempts to address the technical obstacles inherent in visualization of data derived from high-order numerical methods and to provide a robust and easy to use algorithm to overcome the difficulties that arise due to lack of smoothness.
This work is supported in part by the European Office of Aerospace Research and Development (EOARD) under the U.S. Air Force Office of Scientific Research (AFOSR).
Future Accuracy Enhancement research
In addition to these works in progress, I am also exploring other accuracy enhancement techniques. Specifically, I want to generalize the theoretical results of Cockburn et al., Adjerid et al., and Zienkiewich, Zhu for hyperbolic problems, for both DG methods and the usual finite element methods. These components will combine to allow for more efficient use of the finite element methods across a spectrum of disciplines including aerodynamics, aeroacoustics and materials science.