I guess adding u*v*delta*dx, (Where delta is the delta function) to the Left hand side of your problem would be an approximation to having a «source» at the given point.

Wouldn’t u * delta(0.5 , 0.5) * dx be the same as \int_{\mathbf{x}_0- \epsilon }^{\mathbf{x}_0+\epsilon} \mathbf{u} (\mathbf{x}) \ \delta (\mathbf{x-\mathbf{x}}_0)d\mathbf{x} = \mathbf{u} (\mathbf{x}_0 ) where \mathbf{x}_0=(0.5 , 0.5) ?

In that case isn’t inner((u * delta(0.5 , 0.5) * dx) , v )*dx the same as inner( u(0.5 , 0.5), v )*dx? Am I missing something?

I probably wasnt clear enough when i wrote the first post if delta(0.5,0.5) is the spatially varying delta function around 0.5,0.5 that is what you would like yes.