Darcy Flux

We want to calculate the velocity $ u$ and pressure $ p$ on a domain $ \Omega$ solving the Darcy flux problem

\begin{displaymath}\begin{array}{rcl} u\hackscore{i} + \kappa\hackscore{ij} p\ha...
...j} & = & g\hackscore{i} \\ u\hackscore{k,k} & = & f \end{array}\end{displaymath} (107)

with the boundary conditions

\begin{displaymath}\begin{array}{rcl} u\hackscore{i} \; n\hackscore{i} = u^{N}\h...
...\\ p = p^{D} & \mbox{ on } & \Gamma\hackscore{D} \\ \end{array}\end{displaymath} (108)

where $ \Gamma\hackscore{N}$ and $ \Gamma\hackscore{D}$ are a partition of the boundary of $ \Omega$ with $ \Gamma\hackscore{D}$ non empty, $ n\hackscore{i}$ is the outer normal field of the boundary of $ \Omega$, $ u^{N}\hackscore{i}$ and $ p^{D}$ are given functions on $ \Omega$, $ g\hackscore{i}$ and $ f$ are given source terms and $ \kappa\hackscore{ij}$ is the given permability. We assume that $ \kappa\hackscore{ij}$ is symmetric (which is not really required) and positive definite, i.e there are positive constants $ \alpha\hackscore{0}$ and $ \alpha\hackscore{1}$ wich are independent from the location in $ \Omega$ such that

$\displaystyle \alpha\hackscore{0} \; x\hackscore{i} x\hackscore{i} \le \kappa\h...
...core{i} x\hackscore{j} \le \alpha\hackscore{1} \; x\hackscore{i} x\hackscore{i}$ (109)

for all $ x\hackscore{i}$.



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