Thank you Christopher for pointing out the first order approximation.
In general, I was trying implement the BFI (on the boundary) defined by:[tex](Tu,u).[/tex] We approximate this using [tex]u(x)=\sum_{j=1}^N u_j \phi_j(x), [/tex] and [tex](Tu)(x) = \sum_{j=1}^N i\beta_j u_j \phi_j(x).[/tex]
The BFI is equivalent to [tex]\sum_{j=1}^N i\beta_j |u_j|^2,[/tex] where [tex]u_j = (u,\phi_j).[/tex]