CAREER: Primal-Dual Weak Galerkin Finite Element Methods

职业：原始-对偶弱伽辽金有限元方法

• 批准号: 1749707
• 负责人: Chunmei Wang
• 金额: $ 40万
• 依托单位: Texas State University - San Marcos
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2018-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1749707&HistoricalAwards=false
• 关键词: CAREER; Primal; Dual; Weak; Galerkin

Partial differential equations (PDEs) are an important mathematical tool for modeling many scientific problems and the most common approach for solving these equations usually involves numerical methods. The goal of this project is to extend the theory and develop novel numerical methods of one such method known as the PD-WG algorithm. The resulting computational codes will be made publicly available. In addition, the PI will apply the new methods to several problems in biology and physical sciences. One exciting application is the mathematical modeling of ion channels which are proteins with a hole down their middle, and which serve an important function as gatekeepers for cells. The project also has an educational component that integrates the PI's research activities with training of students at all levels, from K-12, undergraduate, to graduate students, with a focus on serving underrepresented minorities.The goal of this project is to develop novel numerical methods through a coupling of the original/primal governing equations and their dual. The resulting numerical methods are known as "Primal-Dual Weak Galerkin (PD-WG) finite element methods". The main research of this CAREER project will be focused on five primary areas: (1) developing robust numerical schemes by using PD-WG techniques for various PDE problems including the elliptic Cauchy problem, convection dominated convection-diffusion equations, and general linear or nonlinear PDEs; (2) establishing stability and mathematical convergence (including superconvergence) theory for the newly developed PD-WG methods by developing new mathematical tools; (3) numerical simulations for biological problems modeled by the nonlinear Poisson-Nernst-Planck (PNP) equations; (4) devising new concepts in numerical PDEs by integrating key ideas of optimization with WG methods; and (5) developing application-oriented software packages using the PD-WG methods.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

偏微分方程（PDE）是一种重要的数学工具，许多科学问题的建模和最常见的方法来解决这些方程通常涉及数值方法。 这个项目的目标是扩展理论和开发新的数值方法的一个这样的方法被称为PD-WG算法。由此产生的计算代码将公开提供。此外，PI将把新方法应用于生物学和物理科学中的几个问题。 一个令人兴奋的应用是离子通道的数学建模，离子通道是中间有一个洞的蛋白质，它作为细胞的看门人发挥着重要作用。该项目还有一个教育部分，将PI的研究活动与从K-12、本科到研究生的各级学生的培训结合起来，重点是为代表性不足的少数群体服务，该项目的目标是通过耦合原始/原始控制方程及其对偶方程来开发新的数值方法。由此产生的数值方法被称为“原始-对偶弱伽辽金（PD-WG）有限元方法”。本CAREER项目的主要研究将集中在五个主要领域：（1）利用PD-WG技术开发各种PDE问题的鲁棒数值格式，包括椭圆柯西问题，对流占优的对流扩散方程，以及一般线性或非线性PDE;（2）建立稳定性和数学收敛性（3）用非线性Poisson-Nernst-Planck（PNP）方程模拟生物学问题的数值模拟;（4）通过将最优化的关键思想与WG方法相结合，在数值偏微分方程中设计新概念;（5）使用PD-WG方法开发面向应用的软件包。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。