I am interested in solving a standard exterior acoustic scattering problem by truncating the whole domain to a bounded domain D, and using approximation (for example, using the first few spherical harmonics ) of the Dirichlet to Neumann map on the boundary of D.

How to define the bilinear form on the boundary of D in Netgen?

Thank you Christopher for pointing out the first order approximation.

In general, I was trying implement the BFI (on the boundary) defined by:tex.[/tex] We approximate this using [tex]u(x)=\sum_{j=1}^N u_j \phi_j(x), [/tex] and tex(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]

I donâ€™t know if this might be useful but I was trying to implement a collocation BEM scheme using NGSolve for a sound soft scattering problem. I think there might be some connection to your problem. I attach a Jupyter Notebook that still is a work in progress.