To model subsurface flows, we investigated a combination of techniques from the fields of mixed-dimensional modeling, reduced order modeling, and trained neural networks. A specific focus was put on the preservation of physical conversation laws at each step of the solving process. As a secondary research theme, we investigated a formulation of Stokes flow which led to a new multi-point vorticity mixed finite element method. We successfully extended these ideas to the Biot equations for poroelasticity, which models the interaction between flow and elasticity in a porous medium. Third, we developed new domain decomposition methods for Stokes-Darcy problems based on flux-mortar methods. Finally, we discovered a connection between the Čech-de-Rham complex and coupled problems such as multi-porosity models.
In parallel with the mathematical aspects of the project, we contributed to an open-source software package where all numerical experiments from the project can be easily reproduced. This code base, called PyGeoN, is designed to construct discretization methods and solvers for mixed-dimensional problems by providing access to the block matrix structure of the problem. The code availability allows for its usage by the scientific community and further developments.
In total, the project has generated seven scientific articles that have been submitted to peer-reviewed journals. Our findings were presented at seven international conferences, including the SIAM Conference on Mathematical and Computational Issues in the Geosciences and Computational Methods in Water Resources. Moreover, we discussed the results with experts in the field at six academic workshops such as SuPreNum: Structure Preserving Numerical Methods and FRAME: Fractured media; numerical methods for fluid flow and mechanics.
