We study approximation techniques for incompressible
flows with heterogeneous
properties. Speci cally, we study two types of phenomena. The first is the flow of a
viscous incompressible fluid through a rigid porous medium, where the permeability
of the medium depends on the pressure. The second is the
ow of a viscous incompressible fluid with variable density. The heterogeneity is the permeability and the
density, respectively.
For the first problem, we propose a finite element discretization and, in the case
where the dependence on the pressure is bounded from above and below, we prove its
convergence to the solution and propose an algorithm to solve the discrete system. In
the case where the dependence is exponential, we propose a splitting scheme which
involves solving only two linear systems.
For the second problem, we introduce a fractional time-stepping scheme which,
as opposed to other existing techniques, requires only the solution of a Poisson equation
for the determination of the pressure. This simpli cation greatly reduces the
computational cost. We prove the stability of first and second order schemes, and
provide error estimates for first order schemes.
For all the introduced discretization schemes we present numerical experiments,
which illustrate their performance on model problems, as well as on realistic ones.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/ETD-TAMU-2010-08-8263 |
| Date | 2010 August 1900 |
| Creators | Salgado Gonzalez, Abner Jonatan |
| Contributors | Guermond, Jean-Luc |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | thesis, text |
| Format | application/pdf |
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