Concluding the Project MiDiROM: Multipoint Mixed Finite Elements and Locally Conservative Deep Learning Methods for Porous Media Problems

Wietse Boon

MOX, Dipartimento di Matematica, Politecnico di Milano

Monday 11th December 2023

11:00:00

Aula Saleri

We reflect on the MSCA-IF project MiDiROM, presenting its successes and challenges over the past two years, and discussing the remaining open questions. The project's main objective was to develop efficient computational methods for modeling processes in fractured porous media, by combining mixed-dimensional modeling with deep learning approaches. This talk focuses on two prominent research themes within MiDiROM. Firstly, we developed multipoint mixed finite element methods for rotation-based formulations of Stokes flow and Biot poroelasticity. By using exact discrete complexes and a carefully selected quadrature rule, we obtained efficient and stable discretization methods using the lowest-order Raviart-Thomas finite element pair. Secondly, we introduced reduced order modeling techniques for Darcy flow systems capable of generating locally mass conservative flux fields. These methods combine concepts from reduced basis methods, exact discrete complexes, and trained neural networks. Mass conservation is then guaranteed through a "spanning tree solve" and appropriate projection operators. To conduct numerical simulations, we simultaneously developed the Python package PyGeoN, in which discrete complexes are explicitly available. The conclusion of the talk will emphasize various capabilities of this open-source software, such as mixed-dimensional differential operators and a polygonal mesh generator using aperiodic monotiles. Contatto: alessio.fumagalli@polimi.it