RUI: Effective multi-method and domain decomposition approaches to solving multiphysics problems of fluid-porous flow with transport

A fluid partly flowing freely and partly filtrating through a porous medium is fundamental in many important environmental and biological phenomena that occur in our daily lives. The development of state of the art computer models for such fluid systems is relevant in multiple areas of national interest. In biotechnology, such a model can be used in studies of groundwater remediation, filtration of blood through arterial walls in the context of optimal treatment of tumors, and the role of bacterial biofilms in wastewater treatment and infectious diseases. In advanced manufacturing, the models can help design improved oil and air filters. To study and predict such processes, it is important to develop accurate and efficient numerical algorithms to solve the systems of underlying mathematical equations. This project will address some of the fundamental difficulties in the development of such algorithms, for example the incompatibilities of the mathematical models in the free and porous regions, an accurate treatment of transitions at the free/porous boundary, and accounting for solute transport and elasticity of the porous structure. This project will support education through undergraduate research experiences. The free flow is often modeled by the incompressible Stokes equations and the porous medium by Darcy’s law, coupled across the interface by the balance of forces, normal velocities, and a slip condition for the tangential velocity. This project aims to develop, analyze, and implement effective multi-method computational models in three dimensions. The main goals of the project are to (i) develop the ability to combine different methods of discretization to effectively model flow in heterogeneous porous media (e.g., using finite element methods) with boundary integral methods that have many advantages in the free flow region, (ii) incorporate the transport of material by the fluid between the two domains using an advection-diffusion equation, and (iii) investigate extensions to poroelastic media using the quasi-static Biot system (which combines the Darcy flow with elasticity of the porous structure). Robust parallel domain decomposition algorithms will be developed and analyzed to provide a theoretical basis for optimal convergence. Novel boundary integral-based methods will be developed for the advection-diffusion equation. To extend to poroelastic media, this project will investigate loosely coupled iterative schemes with time lagging and domain decomposition to solve independently the three problems: the Stokes, Darcy, and elasticity. The stability and accuracy will be studied in order to design algorithms with optimal convergence properties. 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.