Adaptive Discontinuous Galerkin Methods for Partial Differential Equations

研究了偏微分方程间断Galerkin（DG）的超收敛性质.这些方法正在成为计算求解偏微分方程模拟的大规模问题的重要技术。 对于不连续有限元基础，它们可以捕获不连续性，例如，高精度和高效率的双曲系统;简化自适应H-，P-，R-，细化和产生有效的并行解决方案的程序。 研究人员将研究一维和多维空间中双曲问题的DG解和对流扩散问题的局部间断Galerkin（LDG）解的超收敛性质。超收敛现象的几个方面，包括数值通量的影响，稳定计划，网格结构，和阶变化的超收敛性能将被调查。 他们还将开发一个灵活的连续/不连续有限元方法的框架。 超收敛性质的知识将被用来构造简单的和渐近精确的离散化误差和非常准确的函数的后验估计。 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。 事实上，DG和LDG解决方案在两个和三个空间维度的超收敛特性将被用来开发新的离散化误差的后验估计。 研究人员将探索各种策略来构建非常简单和计算效率高的误差估计。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，仍然需要在最快的计算机上花费很长时间。调查人员将开发非常有效、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.