Discontinuous Galerkin Methods for Partial Differential Equations:

不连续伽辽金方法正成为求解偏微分方程的重要方法。使用不连续的有限元基础，它们以高精度和高效率捕获双曲系统中的不连续点；简化自适应网格细化，顺序变化，并导致有效的并行解决程序。研究双曲型系统的DG解和对流扩散及高阶问题的局部不连续Galerkin（LDG）解在一维和多维空间中的超收敛性。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简单且渐近精确的{\it \ a后验}估计离散化误差和非常精确的感兴趣函数。这两者都为自适应解决方案策略提供了有价值的准确性评估和指导。稳定或限制是必要的高阶DG和LDG方法，以消除在不连续和尖锐过渡层附近的杂散振荡。PI将研究几种基于剩余耗散、解矩和ENO方案的稳定策略，目标是发现在特定情况下提供最佳性能的策略。此外，超收敛特性可用于定位不连续和急剧过渡，因此，提供了开发自适应稳定技术的可能性，这种技术只需要在必要时应用，以避免在光滑解区域中不必要的稳定引起的振荡和精度损失。计算机模拟各种学科的复杂和现实问题仍然需要在最快的计算机上花费很长时间。研究者将开发高效、可靠和准确的偏微分方程不连续伽辽金方法，并在数值软件中实现，可用于解决许多关键领域（如能源和环境）中出现的大规模问题。通过后验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.