Multiscale Weak Galerkin Methods for Flows in Highly Heterogeneous Media

多孔介质中的流体流动在许多领域都很重要，包括石油开采和回收、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。在过去的十年里，发展精确、高效和可靠的数值格式来模拟这种流体流动在数学和工程界受到了极大的关注。然而，非均匀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。多孔介质渗流模拟的困难很大程度上是由于从宏观尺度到微观尺度的不同长度尺度的影响。本研究项目旨在开发准确、高效、可靠的渗流数值算法。弱伽辽金有限元方法(WGFEM)将被用来研究高非均质区域、多孔介质和复杂非均质介质中的流动。正在开发的方法预计将大大提高数值分析在现实的科学和工程应用中的效用。该项目旨在开发新的弱Galerkin(WG)有限元方法(FEMS)，该方法在单元构造和网格生成方面具有良好的灵活性，适用于处理非均匀物理参数。此外，可以预见，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行业合作伙伴开展合作。

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.