On Local-Structure-Preserving Discontinuous Galerkin Methods

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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.