On Local-Structure-Preserving Discontinuous Galerkin Methods

高阶间断Galerkin方法广泛应用于航空声学、电磁学、多孔介质中污染物输运、天气预报和图像处理等科学和工程问题的模拟。 这些方法以其在处理复杂几何形状、不同边界条件和不规则网格方面的灵活性而闻名。 在这个项目中，PI将探索这些方法在选择近似函数时的灵活性，这些近似函数保留了精确解的某些重要特征。 该项目将全面涵盖算法开发，分析，实施和局部结构保持间断Galerkin方法的应用，目的是获得新的数值方法，在计算电磁学，计算固体力学和计算流体力学中比现有的方法表现更好。 本研究将为研究生和高年级本科生的培养提供项目。

High-order discontinuous Galerkin methods are widely used in the simulations of many scientific and engineering problems such as aeroacoustics, electromagnetics, transport of contaminants in porous media, weather forecasting, and image processing, among many others. These methods are known for their flexibility in handling complex geometries, different boundary conditions, and irregular meshes. In this project, the PI will explore the flexibility of these methods in choosing the approximating functions that preserve locally certain important features of the exact solution. This project will comprehensively cover the algorithm development, analysis, implementation, and applications of such local-structure-preserving discontinuous Galerkin methods, with the objective of obtaining new numerical methods that perform better than existing ones in computational electromagnetics, computational solid mechanics, and computational fluid dynamics. The proposed research will provide projects for the training of graduate students and advanced undergraduates.