6 Concluding Discussion
The test problems considered in this paper represent two extreme
situations for the application of the boundary element method:
the boundary of the circle has C¥ continuity whereas the
square has only C0 continuity. From the results in tables 2 and
3 it can be seen that the order of convergence is the same for each of
the elements on the two boundaries. This indicates that C0
continuity is sufficient to obtain the optimum order
of convergence for the two methods considered. Constant and hat elements
both give O(N-2) convergence. Guass elements gives O(N-3)
convergence.
The representation of the circle by a polygon is an extreme form
of boundary approximation. A C¥ boundary is replaced by
a C0 boundary. The effect of this is to give an O(N-2)
convergence for each of the elements in the direct method, for the
indirect method the order of convergence seems to be barely O(N-1).
Even though the order is maintained (with respect to the case when the
boundary is exactly represented) in the direct method using
constant and hat elements, the error observed is still 10 to 25
times larger. Thus for the test problem considered, the approximation
to the boundary by a polygon is the dominant factor in the error,
and the effect of the boundary approximation in the indirect method
seems to be particularly damaging.
When quadratically curved elements are used to approximate the
boundary in the circle test problem and constant elements to
approximate the boundary functions in the direct method, the observed
error returns to that of the exact boundary. This shows that under these
circumstances the effect of boundary approximation is not the
dominant factor, the significant error is due to collocation. Thus the
results in section 5 show that the effect of approximating the
boundary may dominate the error in particular circumstances.
In practical situations, the solution in the domain of the partial
differential equation is of interest, rather than the solution on the
boundary. In subsection 5.3, the boundary element solution is
considered for points ranging from the near-field to the far-field.
In these tests the boundary functions were approximated by constant
elements and the results exhibited O(N-2) convergence. For the
exactly-represented circle the relative error stedily reduces into the
far-field whereas, in the other tests, the relative error in the
far-field is almost uniform.
Acknowledgement
The authors are grateful to Dr G. T. Symm and Mr. G. F. Miller of the
National Physical Laboratory for their advice on the subject of this
paper. The authors are indebted to the reviewers for their helpful comments.