the Institute for Theoretical and Engineering Science
Department of Mathematics

University of Houston

Scientific Computing Seminar

Professor V. Girault
Laboratoire Jacques-Louis Lions , Université Pierre et Marie Curie

A Darcy-Forchheimer model$^*$

Thursday, Oct. 26, 2006
3:00 PM- 4:00 PM
Room 634 S&R1

$^*$ This is common work with M. Wheeler, I.C.E.S., University of Texas at Austin.

Abstract: We consider the steady Darcy-Forchheimer flow of a single-phase fluid in a porous medium in a two or three dimensional domain ${\Omega}$ with boundary $\partial {\Omega}$:

$$\frac{\mu}{\varrho}K^{-1}{\boldsymbol u}+ \frac{\beta}{\varrho}\v... ...\boldsymbol u}\vert{\boldsymbol u}+ \nabla\,p = {\bf0}\ {\rm in }\ {\Omega}\,,$$

$${\rm div}\,{\boldsymbol u}= b\ {\rm in }\ {\Omega}\,,$$

$${\boldsymbol u}\cdot {\boldsymbol n}= g\ {\rm on }\ \partial {\Omega}\,,$$

where $\varrho$ is the density of the fluid, $\mu$ its viscosity, $\beta$ a dynamic viscosity, all assumed to be positive constants, $K$ is the permeability tensor, assumed to be uniformly positive definite and bounded, and $b$ and $g$ are given functions satisfying the compatibility condition:

$$\int_{\Omega}b({\boldsymbol x})d{\boldsymbol x}= \int_{\partial {\Omega}} g(\sigma)d\sigma\,.$$

This nonlinear problem is of monotone type. Under mild regularity assumptions on the data $b$ and $g$, several authors have proven that it has a unique weak solution. We propose to solve it numerically with a finite-element method: discontinuous ${I\!\!P}_0^d$ elements for the velocity ${\boldsymbol u}_h$, $d =2$ or $3$, and discontinuous ${I\!\!P}_1$ Crouzeix-Raviart elements for the pressure $p_h$:

$$\forall {\boldsymbol v}_h\,,\,\frac{\mu}{\varrho}\int_{\Omega}(K^... ...l x} + \sum_{T}\int_T \nabla\,p_h\cdot{\boldsymbol v}_h\,d{\boldsymbol x}= 0\,,$$

$$\forall q_h\,,\, \sum_{T}\int_T \nabla\,q_h\cdot{\boldsymbol u}_h... ...t_{\Omega}q_h\,b\,d{\boldsymbol x}+ \int_{\partial {\Omega}} q_h\,g\,d\sigma\,.$$

We prove that this scheme is convergent, again under mild regularity assumptions on the data, and of order one if the exact solution is sufficiently smooth. This non-linear scheme can be solved by a convergent alternating-directions algorithm.

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