From the Navier-Stokes/Brinkman model, a penalization method has been derived by several authors to compute incompressible Navier-Stokes equations around obstacles. In this paper, convergence theorems and error estimates are derived for two kinds of penalization. The first one corresponds to a L 2 penalization inducing a Darcy equation in the solid body, the second one corresponds to a H 1 penalization and induces a Brinkman equation in the body. Numerical tests are performed to confirm the efficiency and accuracy of the method.

Abstract. 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 fault interfaces and the model finally consists in a coupling between a 2D 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 interfaces is derived to solve the set of equations with the additional differential transmission conditions linking both pressure and normal velocity jumps 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 flows in the case of impermeable or partially/highly permeable fractures. The influence of the variation of the aperture of the fractures is also investigated. The numerical solutions of the asymptotic models are validated by comparing them to the solutions of the global Darcy model or to some analytic solutions.Mathematics Subject Classification. 76S05, 74S10, 35J25, 35J20, 65N15.

This study investigates a fictitious domain model for the numerical solution of various incompressible viscous flows. It is based on the so‐called Navier–Stokes/Brinkman and energy equations with discontinuous coefficients all over an auxiliary embedding domain. The solid obstacles or walls are taken into account by a penalty technique. Some volumic control terms are directly introduced in the governing equations in order to prescribe immersed boundary conditions. The implicit numerical scheme, which uses an upwind finite volume method on staggered Cartesian grids, is of second‐order accuracy in time and space. A multigrid local mesh refinement is also implemented, using the multi‐level Zoom Flux Interface Correction (FIC) method, in order to increase the precision where it is needed in the domain. At each time step, some iterations of the augmented Lagrangian method combined with a preconditioned Krylov algorithm allow the divergence‐free velocity and pressure fields be solved for. The tested cases concern external steady or unsteady flows around a circular cylinder, heated or not, and the channel flow behind a backward‐facing step. The numerical results are shown in good agreement with other published numerical or experimental data. Copyright © 2000 John Wiley & Sons, Ltd.