Collaborative Research: Multidimensional couplings for flow and transport in porous media

The project focuses on the study of physical phenomena coupling different spatial scales. The mathematical and numerical analysis of these coupled problems is challenging because of the lack of smoothness in the solution. Applications of these coupled problems are many; for instance the mathematical modeling of blood flow in organs is important in understanding the mechanisms of organ perfusion, embolization and drug delivery.

The project will train graduate students and undergraduate students in computational and applied mathematics. Research outcomes will be published in research journals, online and presented at scientific meetings.

The overall goal of the project is the formulation, analysis and application of discontinuous Galerkin methods for the solution of coupled one-dimensional and three-dimensional flow and transport processes in porous media. The numerical analysis is challenging because weak solutions exhibit a singularity on the line source. The research team will develop and analyze numerical methods for model problems with singular data.

The methods will combine novel efficient time-stepping algorithms with discontinuous finite element methods. The investigators will apply the algorithms to multidimensional couplings in organ and vasculature. The development and validation of physics-based computational models will be guided by imaging of flow and transport.

Neural network based image image segmentation extracts the geometry of the organ and blood vessels. Students will be involved in the research and they will be trained in numerical analysis and scientific computing. Outreach activities will engage high school students with recent developments in computational mathematics.

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.