Darcy Flux

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

$$\begin{array}{rcl} u\hackscore{i} + \kappa\hackscore{ij} p\ha... ...j} & = & g\hackscore{i} \\ u\hackscore{k,k} & = & f \end{array}$$ (133)

with the boundary conditions

$$\begin{array}{rcl} u\hackscore{i} \; n\hackscore{i} = u^{N}\h... ...\\ p = p^{D} & \mbox{ on } & \Gamma\hackscore{D} \\ \end{array}$$ (134)

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 permeability. 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 $\alpha\hackscore{1}$ which are independent from the location in $\Omega$ such that

$$\alpha\hackscore{0} \; x\hackscore{i} x\hackscore{i} \le \kappa\h... ...core{i} x\hackscore{j} \le \alpha\hackscore{1} \; x\hackscore{i} x\hackscore{i}$$ (135)

for all $x\hackscore{i}$ .