5  Results for the Boundary-Value Problem

Results from the application of the direct and indirect methods to the test problem are given in tables I and II. The numerical solution is given at the point (0.25, 0.5) in the domain. The number of elements used in the experiment are 9 (8 boundary and 1 shell), 18 (16 + 2), 36 (32 + 4), 72 (64 + 8), 144 (128 + 16) and 288 (256 + 32) uniform elements. Thus the element lengths h are [1/2], [1/4], [1/8], [1/16], [1/32] and [1/64] respectively.


Table I: Solution via the direct BSEM
h k=1.0 k=2.0 k=4.0
[1/2] 0.5021 + i 0.0014 0.7652 + i 0.0010 -0.7405 + i 0.1209
[1/4] 0.4423 + i 0.0004 0.6589 + i 0.0004 -0.5393 + i 0.0372
[1/8] 0.4157 + i 0.0001 0.6119 + i 0.0001 -0.5077 + i 0.0095
[1/16] 0.4030 + i 0.0000 0.5895 + i 0.0000 -0.5013 + i 0.0024
[1/32] 0.3968 + i 0.0000 0.5785 + i 0.0000 -0.5000 + i 0.0006
[1/64] 0.3938 + i 0.0000 0.5730 + i 0.0000 -0.5000 + i 0.0001


Table II: Solution via the indirect BSEM
h k=1.0 k=2.0 k=4.0
[1/2] 0.4680 + i 0.0055 0.7277 + i 0.0309 -0.8663 + i 0.2701
[1/4] 0.4335 + i 0.0022 0.6454 + i 0.0111 -0.6509 + i 0.2055
[1/8] 0.4120 + i 0.0012 0.6049 + i 0.0057 -0.5775 + i 0.1223
[1/16] 0.4014 + i 0.0007 0.5858 + i 0.0032 -0.5452 + i 0.0743
[1/32] 0.3961 + i 0.0004 0.5764 + i 0.0019 -0.5278 + i 0.0459
[1/64] 0.3934 + i 0.0003 0.5718 + i 0.0012 -0.5174 + i 0.0286


The test problem can be solved using the BEM in the way outlined in the introduction and illustrated in figure 2. Since the solution is antisymmetric about the centre line then the problem is equivalent to the test problem illustrated in figure 4. The direct formulation is now 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, and 192. The results from the application of the standard direct and indirect BEMs are given in tables III and IV.


Table III: Solution via the direct BEM
h k=1.0 k=2.0 k=4.0
[1/2] 0.4825 + i 0.0004 0.6979 - i 0.0055 -0.8005 + i 0.0311
[1/4] 0.4318 + i 0.0003 0.6311 - i 0.0011 -0.5797 + i 0.0100
[1/8] 0.4095 + i 0.0001 0.5983 - i 0.0003 -0.5238 + i 0.0014
[1/16] 0.3997 + i 0.0000 0.5828 - i 0.0001 -0.5071 - i 0.0001
[1/32] 0.3951 + i 0.0000 0.5752 - i 0.0000 -0.5021 - i 0.0002
[1/64] 0.3929 + i 0.0000 0.5714 - i 0.0000 -0.5006 - i 0.0001


Table IV: Solution via the indirect BEM
h k=1.0 k=2.0 k=4.0
[1/2] 0.4582 + i 0.0101 0.6894 + i 0.0377 -0.7778 + i 0.3392
[1/4] 0.4243 + i 0.0037 0.6237 + i 0.0130 -0.6483 + i 0.2111
[1/8] 0.4070 + i 0.0018 0.5943 + i 0.0062 -0.5857 + i 0.1232
[1/16] 0.3988 + i 0.0010 0.5805 + i 0.0034 -0.5512 + i 0.0743
[1/32] 0.3947 + i 0.0006 0.5738 - i 0.0020 -0.5314 + i 0.0457
[1/64] 0.3927 + i 0.0003 0.5706 - i 0.0012 -0.5196 - i 0.0284