Multiscale Weak Galerkin Methods for Flows in Highly Heterogeneous Media

多孔介质中的流体流动在石油开采和回收、环境保护、节能以及燃料电池、太阳能电池和电池的设计和运行等许多领域都具有重要意义。在过去的十年里，开发准确、高效、可靠的数值方案来模拟这种流体的流动受到了数学界和工程界的极大关注。然而，非均质介质和现实环境中流体流动的数学建模和数值模拟仍然是一个挑战。多孔介质流动模拟的许多困难是由于涉及不同的长度尺度，从宏观尺度到微观尺度。本研究项目旨在开发准确、高效、可靠的多孔介质流动数值算法。弱伽辽金有限元方法（WGFEMs）将用于高度非均质区域、多孔介质和非均质介质中的复杂流动。正在开发的方法预计将大大促进数值分析在现实科学和工程应用中的效用。研究生也参与了这个项目。本项目旨在开发新的弱伽辽金（WG）有限元方法，该方法在单元构造和网格生成方面具有良好的灵活性，适合处理异构物理参数。此外，预计新的多尺度wgfem将应用于其他领域，如结构分析、电磁波散射、图像处理和计算机视觉。该研究项目计划与石油工业伙伴合作。

Fluid flow in porous media is important in many areas, including oil extraction and recovery, environmental protection, energy conservation, and the design and operation of fuel cells, solar cells, and batteries. Development of accurate, efficient, and reliable numerical schemes to simulate such fluid flow has received considerable attention in mathematics and engineering communities over the past decade. However, mathematical modeling and numerical simulation of fluid flows in heterogeneous media and realistic settings remain a challenge. Much of the difficulty in porous media flow simulations is due to the involvement of different length scales, from macroscopic scale to microscopic scale. This research project aims to develop accurate, efficient, and reliable numerical algorithms for flows in porous media. Weak Galerkin finite element methods (WGFEMs) will be developed for flows in highly heterogeneous domains, porous media, and complex flows in heterogeneous media. The methods under development are anticipated to significantly advance the utility of numerical analysis for realistic scientific and engineering applications. Graduate students are involved in the project.This project aims to develop new weak Galerkin (WG) finite element methods (FEMs) with excellent flexibility in element construction and mesh generation, suited to dealing with heterogeneous physical parameters. Additionally, it is envisioned that the new multiscale WGFEMs will be applicable in other fields, such as structural analysis, electromagnetic wave scattering, image processing, and computer vision. Collaboration with petroleum industry partners is planned in this research project.