Superconvergent post-processing of some newly developed numerical methods with weak derivatives

近年来，由于科学和工程的需求，计算数学得到了迅速的发展。人们提出并分析了许多逼近偏微分方程解的新方法和新算法。与传统的有限元、有限差分、有限体积法相比，这些新发展起来的方法还处于起步阶段，尤其是在后处理技术方面。在PI和他的学生为有限元方法设计的一种名为多项式保存恢复的特殊后处理技术(PPR自2008年起被商业软件COMSOL多物理采用)的成功的鼓舞下，本项目旨在为其他一些新开发的数值方法开发后处理技术。拟议的项目不仅将设计算法，还将为统一框架下的后处理技术建立数学基础。建议的研究直接应用于其他科学学科，如经典力学、分子动力学、流体动力学、电动力学、等离子体物理、相对论和天文学。该项目的成功将对科学和工程实践以及理论数学的发展产生影响。这个项目的目标是为一些新发展的数值方法，如弱Galerkin方法、虚拟单元方法、杂交间断Galerkin方法等，开发一些健壮的、高精度的后处理算法和相关的数学理论，并致力于开发上述数值方法的与问题和方法无关的梯度恢复技术。将利用连续有限元方法的多项式保持恢复(PPR)的实现和理论，并进一步发展和结合上述数值方法的最新发展的算法和理论。

There has recently been rapid development in computational mathematics due to the demands from science and engineering. Many new methods and algorithms for approximating solutions of partial differential equations have been proposed and analyzed. Compared to traditional methods such as finite element, finite difference, and finite volume methods, these newly developed methods are still in their infancy, especially with respect to post-processing techniques. Encouraged by the success of a special post-processing technique called Polynomial Preserving Recovery (PPR has been adopted by the commercial software COMSOL Multiphysics since 2008) designed by the PI and his students for finite element methods, this project is intended to develop post-processing techniques for some other newly developed numerical methods. The proposed project will not only design algorithms, but also establish a mathematical foundation for post-processing techniques under a unified framework. The proposed research has direct application to other scientific disciplines such as classical mechanics, molecular dynamics, hydrodynamics, electrodynamics, plasma physics, relativity, and astronomy. The success of the project will impact science and engineering practice as well as theoretical mathematical development. The goal of this project is to develop some robust and high accuracy post-processing algorithms and related mathematical theory for some newly developed numerical methods such as Weak Galerkin methods, Virtual Element methods, Hybridizable Discontinuous Galerkin methods, etc. Research efforts will be devoted to developing problem and method independent gradient recovery techniques for the aforementioned numerical methods. The implementation and theory of the Polynomial Preserving Recovery (PPR) for continuous finite element methods will be utilized and further developed and combined with recently developed algorithms and theory for the aforementioned numerical methods.