# [example][bugfix] Change sign of flux in poisson-mfem

Starting from the Poisson equation `-\Delta u = f`

and setting `\sigma = \nabla u`

(or `\sigma - \nabla u =0`

) we obtain
`\nabla \cdot \sigma = -f`

. Using partial integration
we obtain the weak form
`\int \sigma \cdot \tau + u \nabla \cdot \tau= 0`

of the flux equation. Thus the combined weak formulations reads:

`\int \sigma \cdot \tau + u \nabla \cdot \tau + (\nabla \cdot \sigma)v = \int -fv`

.

So far the example contained a minus sign in front of both
flux-pressure terms. This corresponds to defining `\sigma`

to be the negative flux `\sigma = -\nabla u`

. Hence the
essential boundary condition for `\sigma`

was in fact a negative
flux BC. Maybe this was on purpose but I cannot see any reason
for this.

This MR changes the sign, such that the essential BC
for `\sigma`

corresponds to a Neumann BC for `u`

.

It also modifies the boundary values and rhs of the example, because the previous ones where so small that one could not see the sign error.