Development of an Efficient, Parameter Uniform and Robust Fluid Solver in Porous Media with Complex Geometries

The incompressible fluid model is widely used in various fields in engineering and science and their numerical solutions are of prominent importance in understanding complex, natural, engineered, and societal systems. This project aims to advance the state of the art by developing and demonstrating a low-cost, uniform, and parameter robust scheme for fluid flows with mass conservation via a pressure robust finite element method. The proposed research offers significant advancements in accuracy, efficiency, effectiveness, robustness, flexibility, and reliability of simulations for practical applications. This project will strengthen interdisciplinary collaborations among researchers who have different expertise in applied mathematics, computer science, computational fluid dynamics, computational physics, and petroleum engineering. Undergraduate and graduate students will receive interdisciplinary education through course development, research project design, and student training, with a special emphasis on supporting women and underrepresented minority students. The techniques will be implemented in an open-source software package and will be made available to the scientific community. Training of graduate students on the topics of the project is also expected. The proposed mathematical modeling and computational methods address key scientific challenges in the fluid simulation. Novel mathematical formulations are formed based on a divergence preserving numerical scheme and thus guarantee the fundamental mass conservation. The major components include: 1) Designing an applicable scheme that breaks the limitations of traditional solvers to achieve both pressure robustness and mass conservation. 2) Developing a uniform scheme with minimal computational cost, capable of handling varying physical parameters in the mixed regime. 3). Designing a flexible and affordable numerical scheme that can handle problems with complex geometries, providing high grid flexibility. 4). Investigating the robust and effective linear solvers for the resulting discrete linear systems, which is robust with respect to physical and discretization parameters. For all aspects of the project, rigorous numerical analysis will be carried out to prove and validate stability, convergence, and robustness. Corresponding numerical experiments will be performed for further validation. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.