Collocation is applied to derive the boundary element form of
the eigenvalue problems.
The boundary S is divided into n elements
DS1, DS2, ... , DSn and the boundary functions
are approximated by a constant on each element. Let p1, p2,..., pn be the collocation points such that pj Î DSj
for j=1, 2, ..., n. For the methods considered in this paper, DSj
is smooth at pj so that c(pj) = [1/2]. This reduces
the integral equation formulation of the eigenvalue problems to one of
the form (4) where the matrix Ak is defined as follows:
Robin/indirect
[Ak]ij = a(pi) { Lk h }DSj (pi)+ b(pi)
æ è
{Mkt h }DSj (pi) +
1
2
dij
ö ø
,
Dirichlet/indirect or direct
[Ak]ij = {Lk h }DSj (pi) ,
Neumann/indirect
[Ak]ij = {Mkt h }DSj (pi) +
1
2
dij ,
Neumann/direct
[Ak]ij = {Mk h }DSj (pi) +
1
2
dij
where h is the unit function and dij is the Kronecker delta.