8 Concluding Discussion
In this paper a new numerical method termed the boundary and
shell element method has been described and demonstrated on the
two-dimensional discontinuous interior Helmholtz equation. The method is based
on an integral equation that models the boundary of the domain
and discontinuities
directly. For such problems, the standard boundary element method
requires the shell discontinuities to be extended and the
domain divided into several regions, thus requiring a greater
number of elements. Thus the BSEM is generally more efficient than
the standard boundary element method for discontinuous problems.
Tables I and II demonstrate both the direct and
indirect BSEMs on the test problem illustrated in figure 3.
These results are verified by the results of tables
III and IV where the direct and indirect BEMs are applied. The
results for the sample point in each of the tables
are converging to the same solution.
The exact solution is necessarily real, the imaginary parts to the
solution in tables I to IV is error arising through the application
of the method. All of the results
in the tables show O(h) convergence.
The boundary and shell element method has been applied for the
eigenvalue problem associated with figure 3 with homogeneous boundary
conditions.
In tables V and VI the eigenvalues of the discontinuous interior
Helmholtz equation are found by applying the method introduced in
[8] to the boundary and shell element matrices.
The results show convergence as the number
of elements is increased and the
precision of the interpolant is improved. However, as is found with the continuous case [8], the convergence rate does not follow a simple formuls.
In this paper it has been demonstrated that the Boundary and Shell Method is a powerfl tool in the solution of interior Helmhltz problems
with discontinuities in the domain. The aplication of the method to a three-dimensional case - as for the exterior
Helmholtz equation in [17] would also be useful.
Acknowledgement. The work in this paper was partially
funded by SERC/DRA
grant (GR/G01416). The Cray Supercomputer at Rutherford Appleton
Laboratory was used to obtain the results from the test problems
(Supercomputing Grant GR/F61639).