6  Test Problems and Results

In this section some results from the application of the method outlined above to two test problems are presented. In the first test problem the boundary is a sphere and the solution is axisymmetric. In the second test problem the boundary is a square. Both the sphere and square have separable geometries and hence analytic solutions to the Helmholtz eigenvalue problem can be obtained for both cases.
In both tests the boundaries are represented exactly in the boundary element method. The boundary functions are approximated by a constant on each element. The wavenumber range [0,kmax] is divided into equal subintervals in each test. The numerical integrations required in the discretization of the integral operators were carried out with sufficient accuracy to ensure that the significant error in the method is due to the application of collocation and the approximation (9).

6.1  The Sphere Test Problem

For this test, S is a sphere of radius 1.0, centred at the origin. Only axisymmetric solutions are considered. The surface of the sphere is defined by the points ( sin zcos q, sin zsin q, cos z) where ( z, q) Î [ 0 , p] ×[ 0 , 2 p). The surface of the sphere is divided into n axisymmetric elements DS1, DS2, ... , DSn where DSi is the set of points defined by ( z, q) Î [(i-1) p/ n, i p/ n ] ×[ 0 , 2 p), thus the elements are uniform along the generating semicircle, and the relevant boundary functions are approximated by a constant on each element. For the sphere Mk º Mkt and so the resulting matrix approximations to the operators have the same property, hence the results from the application of the direct and indirect boundary element methods to the Neumann problem will be the same. Reference [24] contains further detail on the implementation of this test problem.
In the following tables the approximations to the first four Dirichlet eigenvalues and the first two Neumann eigenvalues are presented. Results are given for each combination of n = 8, 16, 32, kB - kA = 1.0, 0.5 and m = 2, 4. Writing the Helmholtz equation (1) in spherical polar coordinates reduces it to the Bessel differential equation. The Dirichlet eigenvalues are the zeros of Jl(k) , the Bessel function of integer order l, and the Neumann eigenvalues are the zeros of Jl¢(k). The eigenvalues are listed in [31]. To six decimal places, the exact solutions to the Helmholtz eigenvalue problem are k* = 3.141593, 4.493409, 5.763459, 6.283185 for the Dirichlet problem and k* = 2.081576, 3.342094 for the Neumann problem. Computed results are also given to six decimal places


Computed Dirichlet Eigenfrequencies from Lk
n kB - kA m approx. to 3.141593 approx. to 4.493409
8 1.0 2 3.142023 + i0.000511 4.490019 - i0.000293
8 1.0 4 3.141631 + i0.000133 4.490668 - i0.000013
8 0.5 2 3.140540 - i0.000811 4.487145 - i0.000161
8 0.5 4 3.141594 - i0.000001 4.490688 - i0.000007
16 1.0 2 3.142021 + i0.000511 4.492603 - i0.000209
16 1.0 4 3.141628 + i0.000133 4.493086 - i0.000004
16 0.5 2 3.140537 - i0.000811 4.489372 - i0.000150
16 0.5 4 3.141591 - i0.000001 4.493105 + i0.000001
32 1.0 2 3.142020 + i0.000511 4.492900 - i0.000201
32 1.0 4 3.141628 + i0.000133 4.493363 - i0.000004
32 0.5 2 3.140537 - i0.000811 4.489627 - i0.000149
32 0.5 4 3.141591 - i0.000001 4.493382 - i0.000002


Computed Dirichlet Eigenfrequencies from Lk
n kB - kA m approx. to 5.763459 approx. to 6.283185
8 1.0 2 5.757552 - i0.000422 6.274289 - i0.005481
8 1.0 4 5.749567 - i0.000199 6.283160 - i0.000039
8 0.5 2 5.749560 - i0.000205 6.283861 - i0.000131
8 0.5 4 5.749566 - i0.000200 6.283204 - i0.000001
16 1.0 2 5.769463 - i0.000337 6.274270 - i0.005481
16 1.0 4 5.761951 - i0.000001 6.283139 - i0.000038
16 0.5 2 5.762160 + i0.000055 6.283840 + i0.000131
16 0.5 4 5.761967 - i0.000002 6.283184 + i0.000001
32 1.0 2 5.770713 - i0.000344 6.274268 - i0.005481
32 1.0 4 5.763259 + i0.000001 6.283138 - i0.000038
32 0.5 2 5.763491 + i0.000063 6.283839 + i0.000131
32 0.5 4 5.763277 + i0.000000 6.283183 + i0.000001


Computed Neumann Eigenfrequencies from Mk + [1/2] I = Mkt + [1/2] I
n kB - kA m approx. to 2.081576 approx. to 3.342094
8 1.0 2 2.094241 - i0.003572 3.344532 - i0.015619
8 1.0 4 2.084608 - i0.004253 3.351410 - i0.017887
8 0.5 2 2.084234 - i0.003967 3.352193 - i0.018590
8 0.5 4 2.084534 - i0.004218 3.351470 - i0.017949
16 1.0 2 2.092520 - i0.001057 3.337632 - i0.002198
16 1.0 4 2.082453 - i0.001171 3.344606 - i0.004848
16 0.5 2 2.082114 - i0.001004 3.345360 - i0.005531
16 0.5 4 2.082384 - i0.001149 3.344665 - i0.004907
32 1.0 2 2.092043 - i0.000362 3.335682 - i0.001540
32 1.0 4 2.081854 - i0.000316 3.342698 - i0.001208
32 0.5 2 2.081524 - i0.000184 3.343455 - i0.001884
32 0.5 4 2.081787 - i0.000299 3.342758 - i0.001267

6.2  The Square Test Problem

The test problem is that of a square with vertices (0,0), (1,0), (1,1), and (0,1). The interior Helmholtz equation has Dirichlet and Neumann eigenvalues at k* = Ö{j2 + l2} p where j and l are integers, arising when we seek eigenfunctions of the form j(p) = sin(j pp1) sin(l pp2) to the Dirichlet problem or j(p) = cos(j pp1) cos(l pp2) to the Neumann problem. The first three eigenvalues of this form are 4.442883, 7.024815 and 8.885766, though the Neumann problem also has other eigenvalues. For these tests, the matrix components were computed using the Fortran subroutine H2LC, described in reference [32].
In the following tables the approximation to the first three Dirichlet eigenvalues of the above form are given. The boundary of the square is divided into n uniform elements. Results are given for each combination of n=32,64, kB - kA = 2.0, 1.0 and m=2,4. The computed results are given to four decimal places.


Computed Dirichlet Eigenfrequencies from Lk
n kB - kA m approx. to 4.442883 approx. to 7.024815 approx. to 8.885766
32 2.0 2 4.4323 - i0.0154 7.0230 + i0.0001 8.8646 - i0.0065
32 2.0 4 4.4415 + i0.0001 7.0213 - i0.0000 8.8741 - i0.0003
32 1.0 2 4.4405 - i0.0010 7.0269 + i0.0056 8.8733 - i0.0003
32 1.0 4 4.4415 - i0.0000 7.0213 - i0.0000 8.8742 - i0.0001
64 2.0 2 4.4335 - i0.0154 7.0264 + i0.0012 8.8754 - i0.0058
64 2.0 4 4.4427 + i0.0001 7.0244 - i0.0000 8.8843 - i0.0001
64 1.0 2 4.4417 - i0.0010 7.0298 + i0.0054 8.8829 - i0.0002
64 1.0 4 4.4427 - i0.0000 7.0244 - i0.0000 8.8844 - i0.0000


Computed Neumann Eigenfrequencies from Mk + [1/2] I
n kB - kA m approx. to 4.442883 approx. to 7.024815 approx. to 8.885766
32 2.0 2 4.4283 - i0.0082 7.0391 - i0.0228 8.8880 - i0.0378
32 2.0 4 4.4485 - i0.0097 7.0367 - i0.0200 8.8974 - i0.0364
32 1.0 2 4.4471 - i0.0093 7.0433 - i0.0178 8.8971 - i0.0338
32 1.0 4 4.4483 - i0.0097 7.0366 - i0.0199 8.8976 - i0.0365
64 2.0 2 4.4243 - i0.0018 7.0299 - i0.0063 8.8781 - i0.0082
64 2.0 4 4.4444 - i0.0025 7.0277 - i0.0050 8.8882 - i0.0091
64 1.0 2 4.4430 - i0.0018 7.0353 - i0.0038 8.8873 - i0.0079
64 1.0 4 4.4442 - i0.0024 7.0276 - i0.0050 8.8884 - i0.0091


Computed Neumann Eigenfrequencies from Mkt + [1/2] I
n kB - kA m approx. to 4.442883 approx. to 7.024815 approx. to 8.885766
32 2.0 2 4.4783 - i0.0896 7.0907 - i0.1025 8.9358 - i0.1174
32 2.0 4 4.5025 - i0.0960 7.0866 - i0.0928 8.9426 - i0.1097
32 1.0 2 4.5013 - i0.0984 7.0878 - i0.0870 8.9447 - i0.1023
32 1.0 4 4.5024 - i0.0963 7.0866 - i0.0926 8.9428 - i0.1096
64 2.0 2 4.4549 - i0.0517 7.0611 - i0.0575 8.9069 - i0.0588
64 2.0 4 4.4773 - i0.0564 7.0579 - i0.0516 8.9151 - i0.0560
64 1.0 2 4.4760 - i0.0575 7.0622 - i0.0474 8.9150 - i0.0515
64 1.0 4 4.4771 - i0.0566 7.0578 - i0.0515 8.9153 - i0.0556