Multiscale Weak Galerkin Methods for Flows in Highly Heterogeneous Media

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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.