We study the structural stability of a problem in a porous medium when the density of saturating liquid is a nonlinear function of temperature and an internal heat source is present. We prove a convergence result for the Forchheimer coefficient. That is to say, when λ → 0, the solution of the non-isothermal flow in a porous medium of the Forchheimer type, see (1.1), can converge to the solution of the equivalent Darcy type.

We consider the flow of a viscous incompressible fluid through a rigidhomogeneous porous medium. The permeability of the medium dependson the pressure, so that the model is nonlinear. We propose a finiteelement discretization of this problem and, in the case where thedependence on the pressure is bounded from above and below, we proveits convergence to the solution and propose an algorithm to solvethe discrete system. In the case where the dependenceon the pressure is exponential, we propose a splittingscheme which involves solving two linear systems, but parts of the analysis of this method are still heuristic. Numerical tests are presented, which illustrate the introduced methods.


As a first draft of a model for a river flowing on a homogeneous porous ground, we consider a system where the Darcy and Stokes equations are coupled via appropriate matching conditions on the interface. We propose a discretization of this problem which combines the mortar method with standard finite elements, in order to handle separately the flow inside and outside the porous medium. We prove a priori and a posteriori error estimates for the resulting discrete problem. Some numerical experiments confirm the interest of the discretization.

We analyze residual and hierarchicala posteriori error estimates for nonconforming finite elementapproximations of elliptic problems with variable coefficients.We consider a finite volume box scheme equivalent toa nonconforming mixed finite element method in a Petrov–Galerkinsetting. We prove thatall the estimators yield global upper and local lower bounds for the discretizationerror. Finally, we present results illustrating the efficiency of theestimators, for instance, in the simulation of Darcy flows throughheterogeneous porous media.