5  Results from the tests

The measure of error used to estimate the accuracy of the methods is the mean relative error in j
MRE = 1
N
 N
å
j=1 
|
^
j
 
j 
 - jj |
|jj|
  ,
where [^(j)]j and jj are the computed and exact values of j at the collocation points.

5.1  On the presentation of results

The behaviour of the error as the number of elements, and hence the number of collocation points, increases is found to follow a well-defined trend in most cases. This is true for both the direct and to a slightly less extent the indirect method. The evidence for the trends comes from a large amount of data which would be tedious to reproduce in this paper; an interested reader is referred to reference [4]. The method used to condense the data for a sequence of collocation points is shown below.
Consider the data collected from a particular test - the direct method applied to the circle with geometry exactly represented and using constant elements. The results are listed in table 1.

first estimate
N MRE log2 MRE differences using 0.2 ×N-2
4 1.02 ×10-2 -6.62 - 1.25 ×10-2
8 2.26 ×10-3 -8.79 -2.17 3.12 ×10-3
16 7.10 ×10-4 -10.46 -1.67 7.81 ×10-4
32 1.93 ×10-4 -12.34 -1.88 1.95 ×10-4
64 5.03 ×10-5 -14.28 -1.94 4.88 ×10-5
128 1.28 ×10-5 -16.25 -1.97 1.22 ×10-5




The first difference indicates that the errors are approximately proportional to N-2 and using the value for N=128 the constant of proportionality is given as 0.21; thus the error can be taken approximately as 0.2 ×N-2. The final column of table 1 evaluates the corresponding estimates of the MRE. This method of summarising the data has been used in the tables which follow, where the approximations obtained are at least as representative of the data as that shown above.

5.2  Results for the direct method applied to the test problems

In this section the rate of convence that is observed when the direct method is applied to the test problems is given. In table 2 the results for the solution of the Laplace equation with different elements are given for the circle test problem

function geometry approximation
approximation exact polygon quadratically curved
constant 0.2 ×N-2 4.7 ×N-2 0.2 ×N-2
hat 0.4 ×N-2 5.2 ×N-2 no experiment
Gauss 0.3 ×N-3 16.2 ×N-2 no experiment




Remarks
(1) With the geometry exactly represented, the constant and the hat elements give similar results and the same order of convergence. The Gauss elements, although not representing a continuous approximation, give rise to a higher order of convergence, O(N-3).
(2) With the exact geometry of the circle replaced by a polygon the errors for the constant and hat elements are much larger than those observed when the boundary is exactly represented. For the Gauss elements the order of convergence is reduced. This is consistent with the proposition that the effect of the boundary representation on the error is O(N-2).
(3) The previous comment is strengthened by the observation that the accuracy is restored when the polygon is replaced by a quadratically curved geometry for the constant elements. It is likely that a similar result holds for the hat and Gauss elements, though no test was conducted.
For the square test problem the results are summarised in table 3. In all test the boundary is exactly represented.

function approximation direct indirect
constant 4.4 ×N-2 7.8 ×N-2
hat 4.4 ×N-2 no experiment
Gauss 4.5 ×N-3 5.4 ×N-3




Remarks
(1) As in table 2, the constant and hat elements give similar results for the direct method and the Gauss elements give an improved rate of convergence.
(2) Similar results were obtained for both the direct and indirect methods; both give well-defined rates of convergence with the direct method slightly inferior. a further comparison of the two methods is given in the next subsection.

5.3  Comparing the direct and indirect methods

The MRE has been obtained for a number of cases and the results are summarised here and in tables 2 and 3. For the circle test problem the indirect and direct methods are equivalent and they give identical solutions. Replacing the circle by quadratically curved elements and using constant basis functions gives very similar results. The error is characterised as 0.2 ×N-2, as it is in the case when the geometry is represented exactly.
The MRE for constant elements with the boundary approximated by a regular polygon is given in table 4.

direct method indirect method
N MRE log2 MRE MRE log2 MRE
4 2.60 ×10-1 -1.94 1.80 ×10-1 -2.84
8 6.97 ×10-2 -3.84 1.48 ×10-2 -6.07
16 1.80 ×10-2 -5.80 1.98 ×10-2 -5.66
32 4.55 ×10-3 -7.78 1.56 ×10-2 -6.00
64 1.15 ×10-3 -9.77 9.22 ×10-3 -6.76
128 2.87 ×10-4 -11.77 4.96 ×10-3 -7.65




Remark
The rate of convergence for the indirect method is not clearly defined. Initially the indirect method is more accurate but as the number of elements increases the advantage moves to the direct method.
A pattern of behaviour similar to that shown in table 3 is obtained when Gauss elements are used - an unclearly defined convergence rate for the direct method with an initial advantage being lost as N increases.

5.4  Errors in the field solution

So far in this section the errors in the boundary function arising from the primary stage have been considered. In this section the errors away from the boundary, in the domain of the partial differential equation are reported.
In each case the boundary element solution is considered for points ranging from the near-field into the far-field. These are along the outward normal to the boundary and at distances 0.125, 0.25, 0.5, 1.0, 2.0, ..., 512.0, 1024.0 from it. For the circle test problem the actual solution points are (0.0, 4.125), (0.0, 4.25), ..., (0.0, 1028.0) and for the test square the points are (2.0, 4.125), (2.0, 4.25), ..., (2.0,1028.0). Only constant approximations to the boundary functions are considered. In figures 5-8 the symbols °, × and à are plotted for the relative errors for the boundary divided into 32, 64 and 128 elements respectively.
For the circle test problem the errors in the field arising from applying the direct method to the circle with the geometry represented exactly and when approximated by a polygon are shown in figures 5 and 6. The corresponding figures for the direct method are very similar.

Figure 5.



Figure 6.

For the square test problem the field results for the direct method are shown in figure 7 (a similar pattern of behaviour is observed in results for the indirect method).

Figure 7.

It is interesting to question what proportion of the error in the field solution is due to the secondary stage of the BEM and what part is due to the primary stage. In order to investigate this, the secondary stage of the direct method was applied to each test problem, using the exact values on the boundary and at the collocation points. Results for the exact and approximately represented circle and for the square show that the accuracy of the solution is not significantly improved even when use is made of the exact solution at the collocation points. As an example figure 8 may be compared with figure 7.

Figure 8.

Remarks
(1) The direct and indirect methods show the same general behaviour in the way that the initial errors propagate as the field point moves away from the boundary.
(2) The relative error in the exterior solution for the exactly represented circle decreases as the point moves further from the boundary. In the other tests the results suggest that the relative error in the solution in the mid-field to the far-field is almost uniform.
(3) The results in figures 7 and 8 show that for this particular test problem the numerical solution is not significantly improved even when use is made of the exact solution at the collocation points.