# Poisson equation: imposed gradient BC?

Hi all,

I’m slowly transitioning from FEniCS to FEniCSx (although I’m also rusty in FEniCS!) and I’m struggling on a seemingly idiotic thing: how to properly define expression vectors to impose a gradient ?
I’m working on this tutorial, with Neumann BCs where the normal derivative is imposed. I want to tweak it to have a given gradient, and compute myself the dot product of the gradient and the outward normal vector.

So, instead of a weak form with g a known scalar function (normal derivative):

``````inner(g, v) * ds
``````

I want to have a vector-valued g (the chosen gradient) and a weak form like this:

``````inner(dot(g, n), v) * ds
``````

I defined my normal vector as `n = ufl.CellNormal(x)` (with x the SpatialCoordinate of the mesh) and I’m not sure how to define `g`. My current approach is to have a scalar function space (V) and a vector function space (W) to define the gradient, and then I do

``````g = fem.Function(W)
g.interpolate(lambda x: (2*x, 4*x))  # f(x) = x² + 2y², so grad(f) = (2x, 4y)
``````

Doing this leads to this error when the variational form is evaluated:

ERROR:UFL:What do you want cell normal in gdim=2, tdim=2 to be?
[long backtrace]

Any idea how to interpret this ?

Thanks a lot in advance!

This should be a `FacetNormal`, as a cell normal is only defined on manifolds.

Note that you can use the following approach:

``````x = ufl.SpatialCoordinate(mesh)
n = ufl.FacetNormal(mesh)
f = x**2 + 2*x**2
You can simply use `ufl.as_vector((expression_x, expression_y))` if you can express the gradient with spatial coordinates.