A nonconforming rectangular finite element method is proposed to solve a fluid structure interaction problem characterized by the Darcy-Stokes-Brinkman Equation with discontinuous coefficients across the interface of different structures. A uniformly stable mixed finite element together with Nitsche-type matching conditions that automatically adapt to the coupling of different sub-problem combinations are utilized in the discrete algorithm. Compared with other finite element methods in the literature, the new method has some distinguished advantages and features. The Boland-Nicolaides trick is used in proving the inf-sup condition for the multidomain discrete problem. Optimal error estimates are derived for the coupled problem by analyzing the approximation errors and the consistency errors. Numerical examples are also provided to confirm the theoretical results.

This study concerns some asymptotic models used to compute the flow outside and inside fractures in a bidimensional porous medium. The flow is governed by the Darcy law both in the fractures and in the porous matrix with large discontinuities in the permeability tensor. These fractures are supposed to have a small thickness with respect to the macroscopic length scale,so that we can asymptotically reduce them to immersed polygonal faultinterfaces and the model finally consists in a coupling between a2D elliptic problem and a 1D equation on the sharp interfaces modelling the fractures.A cell-centered finite volume scheme on general polygonal meshes fitting the interfacesis derived to solve the set of equations with the additional differential transmission conditions linking both pressure and normal velocityjumps through the interfaces.We prove the convergence of the FV scheme for any set of data and parameters of the models and derive existence and uniqueness of the solution to the asymptotic models proposed. The models are then numerically experimented for highly or partially immersed fractures.Some numerical results are reported showing different kinds of flowsin the case of impermeable or partially/highly permeable fractures. The influence of the variation of the aperture of the fractures is also investigated. The numericalsolutions of the asymptotic models are validated by comparing them to the solutions of the global Darcy model or to some analytic solutions.