Exact form on the boundary and Dirichlet approximation

It is possible to calculate $G$ exactly using $O(NM)$ basis function (or derivative) evaluations. The equivalent expressions (H.14) and (H.7) both give the domain overlap of two Helmholtz solutions (with no particular BCs) at the same wavenumber, purely in terms of boundary integrals. Taking for example the former expression, it can be approximated (using the discretization of Appendix G) as

$$G \; = \; \frac{{\mathsf{A}}}{M} \, \frac{1}{2k^2} \left[ ... ... - (A^{{\mbox{\tiny T}}} B + B^{{\mbox{\tiny T}}} A) \right] ,$$ (5.20)

where the four rectangular matrices are defined by $A_{in} = p_i \phi_n({\mathbf r}_i)$, $B_{in} = p_i^{-1} {\mathbf n}_i \cdot (\nabla \nabla \phi_n)({\mathbf r}_i) \cdot {\mathbf r}_i$, $C_{in} = p_i \partial_x \phi_n({\mathbf r}_i)$, and $D_{in} = p_i \partial_y \phi_n({\mathbf r}_i)$. At each boundary point ${\mathbf r}_i$ for $i=1\cdots M$, the weight is $p_i \equiv ({\mathbf r}_i \cdot {\mathbf n}_i)^{1/2}$. The outward normal at this point is ${\mathbf n}_i$. Notice that the (Hessian) matrix of second derivatives is required for $B$.

This formidable expression has been coded and tested. It has been found that for the Dirichlet eigenproblem, it can be replaced by a much simpler form which is valid only when the wavefunctions in question vanish on the boundary, namely (H.9). When written in a basis this gives

$$G_{ij} \; = \; \frac{1}{2k^2} \oint_\Gamma \!\! d{\mathbf s} \,r_n \, (\partial_n \phi_i) \partial_n \phi_j ,$$ (5.21)

where $\partial_n$ means derivative in the normal direction. The discretized version is therefore

$$G \; = \; \frac{{\mathsf{A}}}{M} \, \frac{1}{2k^2} C^{{\mbox{\tiny T}}} C ,$$ (5.22)

where $C_{in} = p_i \partial_n \phi_n({\mathbf r}_i)$. The largest few resulting generalized eigenvalues $\lambda$ (and eigenfunctions) of (5.14) are indistiguishable from those resulting from (5.20). That this works is not entirely obvious, because the basis functions $\phi_n$ do not vanish on the boundary. However all the relevant vectors ${\mathbf x}$ contain combinations of basis functions which are very small on the boundary. For the tension plots in this chapter, (5.22) has been used.

As a bonus, the eigenstates with Neumann BCs are to be found at the minima of the smallest $\lambda$. This is true because exchanging Dirichlet for Neumann in the tension function definition is equivalent to swapping $F$ and $G$ in (5.14).