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

该项目的目标是开发计算方法，用于解决轴对称聚变反应堆磁平衡研究中产生的一族非线性和积分微分方程。作为该项目的一部分开发的工具和技术将有助于对磁约束聚变行为的理论理解。此外，它还将为等离子体物理界提供一种工具，用于有效模拟反应堆中发生的某些过程。因此，该项目有可能促进清洁能源的开发。作为增加少数民族参与科学、技术、工程和数学的更广泛目标的一部分，该项目将为亚利桑那大学的本科生和研究生举办一次座谈会。座谈会将有来自代表性不足的社区的数学家，他们是各种环境（学术界，国家实验室，工业研发和政府机构）中数学的成功实践者。演讲者将提出一个广泛的和可访问的观点，他们的研究，并将与学生互动，分享有关数学专业和自己的教育生涯的建议和个人经验。这将服务于双重目的：（1）通过展示具有不同文化和种族背景的个人的贡献来挑战专业数学家的“传统形象”，以及（2）为学生提供几个非标准的角色模型和更多样化的数学专业形象。该项目的科学部分有两个主要目标：（1）设计和分析广泛的一类非线性椭圆积分微分方程的可杂交不连续Galerkin（HDG）离散化，以及（2）将从上一点获得的结果应用到可直接由计算等离子体物理社区使用的免费软件实现中。考虑的问题的数学兴趣远远超出了等离子体物理领域，并将激励扩展的HDG方法的理论到领域，它没有被应用之前。从应用的角度来看，该项目将开发一套免费的计算工具和软件，将直接影响磁聚变社区，并可以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，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。

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.