Adaptive Discontinuous Galerkin Methods for Partial Differential Equations

研究偏微分方程间断Galerkin(DG)的超收敛性。这些方法正在成为计算求解由偏微分方程组模拟的大规模问题的重要技术。利用不连续的有限元基，它们以高精度和高效率捕捉双曲型系统中的间断；简化自适应h-、p-、r-细化并产生高效的并行求解程序。研究人员将研究一维和多维空间中双曲型问题的DG解和对流扩散问题的局部间断Galerkin(LDG)解的超收敛性质。研究了超收敛现象的几个方面，包括数值通量、稳定化格式、网格结构和阶数变化对超收敛性质的影响。他们还将开发一个灵活的连续/不连续有限元方法的框架。关于超收敛性质的知识将被用来构造离散化误差和非常精确的感兴趣函数的简单且渐近精确的后验估计。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。实际上，DG和LDG解在二维和三维空间的超收敛性质将被用来发展离散化误差的新的后验估计。研究人员将探索各种策略，以构建非常简单和计算高效的误差估计。许多实际的计算机模拟，如汽车或飞机周围的流动或天气预报，在速度最快的计算机上仍然需要很长的时间。调查人员将开发非常高效、可靠和准确的方法。这项工作将导致更准确的计算机模拟，更好的产品和更短的设计周期。

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.