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

一些新发展的弱导数数值方法的超收敛后处理

• 批准号: 1419040
• 负责人: Zhimin Zhang
• 金额: $ 17万
• 依托单位: Wayne State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-15 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1419040&HistoricalAwards=false
• 关键词: Superconvergent; post; processing; some; newly

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.

由于科学和工程的需求，计算数学最近得到了快速发展。人们提出并分析了许多用于逼近偏微分方程解的新方法和算法。与有限元、有限差分、有限体积等传统方法相比，这些新发展的方法仍处于起步阶段，特别是在后处理技术方面。受到 PI 和他的学生为有限元方法设计的称为“多项式保存恢复”（PPR 自 2008 年以来已被商业软件 COMSOL Multiphysics 采用）的特殊后处理技术成功的鼓舞，该项目旨在为其他一些新开发的数值方法开发后处理技术。该项目不仅将设计算法，还将在统一框架下为后处理技术奠定数学基础。拟议的研究可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、等离子体物理学、相对论和天文学。该项目的成功将影响科学和工程实践以及理论数学的发展。该项目的目标是为一些新开发的数值方法（如弱伽辽金方法、虚拟元素方法、可混合非连续伽辽金方法等）开发一些鲁棒且高精度的后处理算法和相关数学理论。研究工作将致力于为上述数值方法开发与问题和方法无关的梯度恢复技术。连续有限元方法的多项式保持恢复（PPR）的实现和理论将得到利用和进一步发展，并与最近开发的上述数值方法的算法和理论相结合。