LEAPS-MPS: Hybridizable discontinuous Galerkin methods for non-linear integro-differential boundary value problems in magnetic plasma confinement

该项目的目标是发展一族非线性积分-微分方程解的计算方法，这些方程是由轴对称聚变堆中的磁平衡研究产生的。作为该项目的一部分开发的工具和技术将有助于从理论上理解磁约束聚变的行为。此外，它还将为等离子体物理界提供一个工具，用于有效地模拟反应堆中发生的一些过程。因此，该项目有可能为清洁能源的发展做出贡献。作为增加少数民族参与科学、技术、工程和数学这一更广泛目标的一部分，该项目将在亚利桑那大学组织一场面向本科生和初级研究生的座谈会。座谈会将有来自代表性不足社区的数学家参加，他们是各种环境(学术界、国家实验室、工业研究和开发以及政府机构)中成功的数学实践者。演讲者将介绍他们研究的广泛和易懂的观点，并将与学生互动，分享与数学专业和他们自己的教育事业有关的建议和个人经验。这将起到双重作用：(1)通过展示具有不同文化和种族背景的个人的贡献，挑战专业数学家的“传统形象”；(2)为学生提供几个非标准的榜样和更多样化的数学职业形象。该项目的科学部分有两个主要目标：(1)设计和分析一类广泛的非线性椭圆型积分-微分方程组的杂交间断Galerkin(HDG)离散化，以及(2)将从上一点获得的结果应用到可供计算等离子体物理界直接使用的免费软件实现中。对所审议问题的数学兴趣远远超出了等离子体物理领域，并将推动HDG方法的理论扩展到以前从未应用过的领域。从应用的角度来看，该项目将开发一套免费提供的计算工具和软件，这些工具和软件将直接影响磁聚变社区，并可有益于应用科学和工程的其他领域。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。

The goal of this project is to develop computational methods for the solution of a family of non-linear and integro-differential equations that arise from the study of magnetic equilibrium in axisymmetric fusion reactors. The tools and techniques developed as part of the project will contribute to the theoretical understanding of the behavior of magnetic confinement fusion. In addition, it will provide a tool to the plasma physics community for the efficient simulation of some of the processes taking place in a reactor. Thus, the project has the potential to contribute to the development of clean energy sources. As part of the broader goal of increasing the participation of ethnic minorities in Science, Technology, Engineering and Mathematics, the project will organize a colloquium addressed to undergraduate and beginning graduate students at the University of Arizona. The colloquium will feature mathematicians from underrepresented communities who are successful practitioners of mathematics in a variety of settings (academia, national laboratories, industrial research and development, and government agencies). Speakers will present a broad and accessible view of their research and will interact with the students, share advice and personal experiences related to the mathematical profession and their own educational career. This will serve the dual purpose of (1) challenging the perceived "traditional image" of a professional mathematician by showcasing the contributions of individuals with diverse cultural and ethnical backgrounds, and (2) providing students with several non-standard role models and a more diverse image of the mathematical profession. The scientific component of the project has two main goals: (1) devise and analyze hybridizable discontinuous Galerkin (HDG) discretizations for a broad class of non-linear elliptic integro-differential equations, and (2) apply the results obtained from the previous point into a freely available software implementation that can be used directly by the computational plasma physics community. The mathematical interest in the problems considered extends well beyond the field of plasma physics and will motivate the expansion of the theory of the HDG method into areas where it had not been applied before. From the point of view of applications, the project will develop a set of freely available computational tools and software that will directly influence the magnetic fusion community and can be beneficial to other areas of applied science and engineering.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.