I am studying the acoustic scattering simulation. I found that the assembled matrix is asymmetric for Bempp. But it is strange because the equation solved in [BEM for Acoustic scattering ]((7) (PDF) Fast Methods applied to BEM Solvers for Acoustic Propagation Problems (researchgate.net)) is similar in Bempp, but the matrix in the literature is dense complex symmtric while the matrix in Bempp is asymmtric. I am wondering what causes the asymmetries in Bempp, the derivative of Green function or other sources?

PS: the description in the attached literature is captured as below:

Dear Long, the developers will know better than me for definite, but I think this could come from the quadrature scheme, or more exactly, the regularisation process for the singular integrals. I implemented HYENA the scheme that bempp uses (or used to use at least) some years ago and it was asymmetrical.

This will only explain asymmetry for self-interaction or near neighbours, however. If youâ€™re getting it in other places then it might be your choice of approximation spaces. For example, if youâ€™re looking at the discretised weak-form boundary operator, then your dual-to-range space will need to match your range space in order to achieve symmetry. If you go for the strong-form instead, then there is a matrix solution going on internally, and even more possible reasons for asymmetry could arise.

Hi, Jonathan, thanks for your sharing. That is helpful to me. I also think it is caused by quadrature scheme in self-interaction or near neighbours. But I am puzzled for this, because if this is the fact, it would suggest that the asymmetry is caused by the numerical issue not the internal asymmetry from the integral operators. In my experiment, I found that if the wavenumber is low (O(1)), the deviation - (L - U.T) is small (O(10^-2)), but it is still symmetric, while the deviation is very large (O(10^2)) for large wavenumber (O(10)). The pheneomenon makes me believe that it is mainly caused by the internal asymmetric struture of the integral operators, and the influence of numerical integration is minor.

I guess that the internal asymmetry is caused by the derivatives of Green function, because the derivative is anisotropic?

The single-layer potential is symmetric, and the Hypersingular operator, called D above, should be too, because the spatial derivatives are applied at both x and y. The double-layer potential and its adjoint are not symmetrical, however, because the derivative is only applied at x or y.

I believe that bempp evaluates the hypersingular operator indirectly via the identity of Mariem and Hamdi (1987), but that is symmetrical too, so shouldnâ€™t be the cause of the issue you are reportingâ€¦

Thanks for your further explanation, Jon. You are correct. I have further verified the operators just now. Hypersingular operator is definitely symmetric (actually weakly nonsymmetric numerically). Double-layler and single layer operators are highly nonsymmetric. Next, let me verify the solution correction if only the hypersingular operator is adopted to construct the equation. If the solution uniqueness is not a big problem, the conjugate gradient method or LDL^T is more effective to solve the discretized linear equations. Besides, the B-M equation has been well verified.