5.3.2 Dirichlet Type (Implicit Flux)

In case an expression $\nu_{i}^{}$ = h($\nu_{i'}^{}$) is available F_{$\nu_{i,i'}$} can be eliminated by adding (5.9) to (5.10) and the resulting equations are of the following form:

$$\nu_{i} \nu_{i}^{} \nu_{i'}^{}$$ = 0 (5.13)

$$\nu_{i'} \nu_{i} \nu_{i'}$$ = 0 (5.14)

As (5.13) is normally not diagonal-dominant it is eliminated in a pre-pass. It is important to note, that in this case the structure of the equation system changes. The constitutive relation for $\nu_{i}^{}$ is now given by (5.13) whereas for $\nu_{i'}^{}$ by (5.14), or vice-versa. This is accomplished by the following transformation matrix

tx, y	$\nu_{i}^{}$	$\nu_{i'}^{}$
$\nu_{i}^{}$
$\nu_{i'}^{}$	1	1

The fluxes contained in f_{$\nu_{i}$} can also be used to calculate the total interface flux F_I by setting up the following equation:

$$\nu_{F_{I}} \sum_{i}^{} \nu_{i}$$ (5.15)

with i running over all interface points.