WS, Anna Scotti: N-1 dimensional interfaces: mixed formulation and discretization methods

Speaker
Anna Scotti,Politecnico di Milano

Abstract
We present an overview on mixed dimensional problems arising from the dimensional reduction of embedded subdomains, focusing on elliptic problems in mixed form . This is the case, for instance, of the Darcy problem in the presence of fractures, [1], or the electrostatic problem with a thin heterogeneous layers, [2]. The co-dimension subdomain is not merely an interface, but we solve a reduced PDE coupled with the surrounding ”bulk” problem. In this talk we outline the strategy to derive such reduced problem, [3], discussing the most common assumptions and the treatment of coupling conditions in mixed form, and then proceed to discuss discretization strategies with their advantages and limitations: the choice of conforming or
non-conforming grids and the monolithic versus iterative solution of the resulting linear system [4]. Finally, we discuss convergence in the presence of modeling error and numerical error, and present some applications to real life problems.

References
[1] D’Angelo, C., Scotti A., A mixed finite element method for Darcy flow in fractured porous media with non-matching grids , ESAIM: M2AN 46 (2012)
[2] Fumagalli A., Panzeri L., Formaggia L., Scotti A., Arosio D., A mixed-dimensional model for direct current simulations in the presence of a thin high-resistivity liner, Int J Numer Methods Eng. 2023;e7407 (2023)
[3] Martin V., Jaffré J., Roberts J.E., A mixed-dimensional model for direct current simulations in the presence of a thin high-resistivity liner, SIAM J. SCI. COMPUT. Vol. 26, No. 5 (2005)
[4] Berrone S., D’Auria A., Scialò S., An optimization approach for flow simulations in poro-fractured media with complex geometries, Computational Geosciences 25:897–910 (2021)