Fall 2008, Thursdays, Milner 216, 4:00-4:50 PM

Date: September 25

"  Linear and Nonlinear Darcy Equations. "
  Abner Salgado  


Around 150 years ago, on the basis of experimental observations, H.Darcy came up with the system of equations that bear his name. These equations are used to model the flow of a viscous incompressible fluid through a porous medium. From the mathematical point of view, these equations are a prototypical example of mixed type problems, saddle point problems and constrained optimization problems. Moreover, their numerical approximation can serve as an excellent example to introduce a fundamental concept in numerical analysis, the so-called discrete inf-sup conditions.

Many nonlinear variants of these equations have been proposed to model different situations. One that is very important (specially considering the currently exorbitant gas prices) is used in the enhanced oil recovery problem. In this case, the porosity is assumed to depend on the pressure of the fluid.

In this talk I will introduce the Darcy equations, show that this problem is well-posed in the sense of Hadamard, and discuss some approximation techniques for these equations. We will see the importance of the discrete inf-sup conditions and present some techniques for solving the linear system of equations to which the approximate problem reduces. If time allows, I will also discuss the above mentioned nonlinear system of equations, show that this system always has a solution, and give sufficient conditions for this solution to be unique. As it is possible for this problem to have more than one solution, we will discuss sufficient conditions for the solution to be locally unique (i.e. isolated or nonsingular.) We will present also the numerical analysis of this problem, for both cases when the solution is globally or locally unique.

I will try to make this talk as accessible as possible and if I am not successful in doing so constructive criticism will be gladly accepted. Therefore, all you basically need to know to get this talk is what a partial derivative is, and maybe the definition of a Banach space and the L^2 space. Of course, I will try to make the talk entertaining for those who know more than that, trying to draw connections to more interesting results.