Discontinuous Galerkin Methods for Partial Differential Equations:

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Discontinuous Galerkin (DG) methods are becoming important techniques for the computational solution of partial differential equations. With discontinuous finite element bases, they capture discontinuities in, {\it e.g.}, hyperbolic systems with high accuracy and efficiency; simplify adaptive mesh refinement, order-variation and lead to efficient parallel solution procedures.The PI proposes to study the superconvergence properties of DG solutions of hyperbolic systems and local discontinuous Galerkin(LDG) solutions of convection-diffusion and higher-order problems in one and multiple space dimensions. The PI will investigate several aspects of the superconvergence phenomena, including the effects of numerical fluxes, stabilization schemes, mesh structure, and order variation on superconvergence properties. The PI will use superconvergence properties to construct simple and asymptotically exact {\it a posteriori} estimates of discretization errors and very accurate functions of interest. Both of these provide valuable accuracy appraisals and guidance for an adaptive solution strategy.Stabilization or limiting is necessary with high-order DG and LDG methods to remove spurious oscillations near discontinuities and sharp transition layers. The PI will investigate several stabilization strategies based on residual dissipation, solution moments, and ENO schemes with a goal of discovering those that provide optimal performance in specified circumstances. Furthermore, superconvergence properties can be used to locate discontinuities and sharp transitions and, as such, provide the possibility of developing adaptive stabilization techniques that need only be applied where necessary to avoid oscillations and loss of accuracy caused by unnecessary stabilization in smooth solution regions.Computer simulations of complex and realistic problems from a variety of disciplines still require a very long time on the fastest available computers. Efficient, reliable and accurate discontinuous Galerkin methods for partial differential equations will be developed by the investigator and implemented in numerical software that can be used to address large-scale problems arising in many critical areas such as energy and environment. The reliability provided through {\it a posteriori} error estimation, will additionally enable advanced adaptive software to be used in educational settings to help students understand delicate and intricate phenomena and prepare the next generation of scientists.