6 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 interior Laplace problem. Similar equations and
methods can be derived for other linear elliptic problems
through appropriate change of Green's function,
in exterior as well as interior domains and are applicable to three-dimensional
problems as well as two.
The BSEM is a useful generalisation of the standard BEM.
For the application of the standard BEM, in the way illustrated in figure 2,
will in many cases result in an inefficient method when the
shells G¢ need to be large with respect to the size of S and G.
Morover the way in which G¢ is joined to the boundary S could
result in numerical problems with the overall method.
Tables 1 and 2 demonstrate both the direct and
indirect BSEMs. These results are verified by the results of tables
3 and 4 where the direct and indirect BEMs are applied. For the
results for each of the three sample points in each of the tables
seem to be converging to the same solution. All of the results
in the tables also clearly indicate
O(h) convergence.
Further results from test problems are given in Kirkup (1990) and the
BSEM for the exterior Helmholtz equation is considered in Kirkup (1991).
Acknowledgement. The work in this paper was funded by SERC/MOD
grant (GR/G01416). The Cray Supercomputer at Rutherford Appleton
Laboratory was used to obtain the results from the test problems
(Supercomputing Grant GR/F61639).