CAREER: Primal-Dual Weak Galerkin Finite Element Methods

偏微分方程(PDE)是模拟许多科学问题的重要数学工具，而求解这些方程的最常用方法通常是数值方法。这个项目的目标是扩展理论并开发出一种新的数值方法，即PD-WG算法。由此产生的计算代码将公开提供。此外，PI将把新方法应用到生物和物理科学的几个问题上。一个令人兴奋的应用是对离子通道的数学建模，离子通道是中间有一个洞的蛋白质，它扮演着细胞守门人的重要功能。该项目还有一个教育部分，将国际和平研究所的研究活动与各级学生的培训结合起来，从K-12、本科生到研究生，重点是服务于代表性不足的少数群体。该项目的目标是通过耦合原始/原始控制方程及其对偶来开发新的数值方法。由此产生的数值方法称为“原始-对偶弱Galerkin(PD-WG)有限元方法”。该项目的主要研究将集中在五个主要方面：(1)利用PD-WG技术开发各种偏微分方程组的稳健数值格式，包括椭圆型柯西问题、对流占优对流扩散方程和一般的线性或非线性偏微分方程组；(2)通过开发新的数学工具，为新发展的PD-WG方法建立稳定性和数学收敛(包括超收敛)理论；(3)用非线性Poisson-Nernst-Planck(PNP)方程模拟生物问题；(4)将优化的关键思想与WG方法相结合，设计数值偏微分方程组的新概念；以及(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智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.