Adaptive Discontinuous Galerkin Methods for Partial Differential Equations

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The investigators intend to study the superconvergence properties of Discontinuous Galerkin (DG) for partial differential equations. These methods are becoming important techniques for the computational solution of large scale problems modeled by partial differential equations. With discontinuous finite element bases, they capture discontinuities in, e.g., hyperbolic systems with high accuracy and efficiency; simplify adaptive h-, p-, r-, refinement and produce efficient parallel solution procedures. The investigators will study the superconvergence properties of DG solutions of hyperbolic problems and local discontinuous Galerkin (LDG) solutions of convection-diffusion problems in one and multiple space dimensions. Several aspects of the superconvergence phenomena, including the effects of numerical fluxes, stabilization schemes, mesh structure, and order variation on superconvergence properties will be investigated. They will also develop a framework for flexible continuous/discontinuous finite element methods. A knowledge of superconvergence properties will be used to construct simple and asymptotically exact 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. Indeed, superconvergence properties of DG and LDG solutions in two and three spatial dimensions will be used to develop new a posteriori estimates of discretization errors. The investigators will explore various strategies to construct very simple and computationally efficient error estimates. Many practical computer simulations such as flow around a car or a plane or weather forcasting still require a very long time on the the fastest computers. The investigators will develop very efficient reliable and accurate methods. This work will lead to more accurate computer simulations, better product and shorter design cycles.