5  Solution of the test problem by the BEM

The test problem applied in the previous section can be solved using the BEM in the way outlined in the introduction and illustrated in figure 2. Moreover, this particular problem is antisymmetric about the centre line. Hence the test problem illustrated in figure 3 is equivalent to the test problem illustrated in figure 4. The direct formulation is the Green's formula, the indirect formulation is the one arising through writing j as a single layer potential. The boundary functions are discretised in a similar way to the method in the previous section. The number of elements in the experiments are 6, 12, 24, 48, 96, 192, and 384 of uniform size h. The results from the application of the standard direct and indirect BEMs are given in tables 3 and 4.




Table 3: Solution via the direct BEM
h p = (0.25,0.25) p = (0.25,0.5) p = (0.25,0.75)
[1/2] 0.211877 0.433832 0.715219
[1/4] 0.192590 0.387755 0.687880
[1/8] 0.184626 0.368262 0.673894
[1/16] 0.180721 0.359744 0.667388
[1/32] 0.178799 0.355801 0.664339
[1/64] 0.177850 0.353914 0.662888
[1/128] 0.177379 0.352993 0.662187


Table 4: Solution via the indirect BEM
h p = (0.25,0.25) p = (0.25,0.5) p = (0.25,0.75)
[1/2] 0.192122 0.397033 0.662352
[1/4] 0.181022 0.375855 0.675031
[1/8] 0.178861 0.363480 0.669478
[1/16] 0.177494 0.357531 0.665706
[1/32] 0.176867 0.354643 0.663606
[1/64] 0.176658 0.353252 0.662522
[1/128] 0.176634 0.352595 0.661985