It is suggested in Kress [13] and Amini [15] that
the weighting parameter should be chosen as a function of k and S only.
For Neumann problems, it has been shown in this paper that
the value chosen for the weighting parameter cannot be independent of the
number of elements if the resulting numerical method is to be consistent.
Figures 4 and 5 show that for the particular test problem used in this
work, choosing a value of parameter that balances the contribution from
the component matrices gives rise to a well-conditioned composite
matrix and generally smaller error.
It is clear from the results from the test problems and the
analysis that the value assigned to the parameter has a major influence
on the resulting error.
However, the error in the test problems is not critically dependent
on l. Thus, given the trade off between the computational cost of
choosing a value for the parameter and its beneficial effect on the
numerical solution, the most efficient route is likely to be that of
making only a rough estimate of its optimal value.
For the exterior Helmholtz equation with a Robin boundary condition
(1), an indirect BEM can be derived in the standard way from the
following integral equation:
f(p) = a(p)
æ è
a{Lk s}S (p) + b{ Mk s} S (p) +
1
2
bs(p)
ö ø
+ b(p)
æ è
a{ Mkt s}S (p) + b{ Nk s}S (p) -
1
2
as(p)
ö ø
forp Î S
which follows from the substitution of (2), (3) into (1).
Derivation of the direct BEM for the Robin problem is more awkward. Nevertheless,
both methods require solution over the Nk integral operator and hence
the outcome of the analysis in this paper for the Neumann problem
will also apply to the more general Robin problem.
Acknowledgement.
The author is grateful to Dr D J Henwood of the Department of
Mathematical Sciences, Brighton Polytechnic for his support in this
work and to Dr G T Symm
and Mr G F Miller of the National Physical Laboratory,
Teddington for their advice on the work in this paper.