Discontinuous Galerkin Methods for Partial Differential Equations:

间断Galerkin（DG）方法是求解偏微分方程的重要方法。 对于不连续的有限元基，他们捕捉不连续性，PI提出研究一维和多维空间中双曲方程组DG解和对流扩散及高阶问题的局部间断Galerkin（LDG）解的超收敛性质。PI将研究超收敛现象的几个方面，包括数值通量，稳定方案，网格结构和阶数变化对超收敛性质的影响。 PI将使用超收敛性质来构造离散化误差和非常精确的函数的简单且渐近精确的{\it a posteriori}估计。这两种方法为自适应求解策略提供了有价值的精度评估和指导。稳定或限制对于高阶DG和LDG方法是必要的，以消除不连续性和尖锐过渡层附近的寄生振荡。 PI将研究基于剩余耗散、解矩和ENO方案的几种稳定策略，目的是发现在特定情况下提供最佳性能的稳定策略。此外，超收敛性质可用于定位不连续性和急剧转变，因此，提供了开发自适应稳定技术的可能性，这些技术只需要在必要时应用，以避免在光滑解区域中由不必要的稳定引起的振荡和精度损失。来自各种学科的复杂和现实问题的计算机模拟仍然需要在最快的现有技术上花费很长时间。电脑研究人员将为偏微分方程开发高效、可靠和准确的间断伽辽金方法，并在数值软件中实施，可用于解决能源和环境等许多关键领域出现的大规模问题。通过误差估计提供的可靠性，还将使先进的自适应软件能够用于教育环境，帮助学生理解微妙和复杂的现象，并为下一代科学家做好准备。

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.