Geodesic
Potential-based numerical solution of Dirichlet problems for the Helmholtz equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 8, pp. 1492-1505

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Three-dimensional Dirichlet problems for the Helmholtz equation are considered in generalized formulations. By applying single-layer potentials, they are reduced to Fredholm boundary integral equations of the first kind. The equations are discretized using a special averaging method for integral operators with weak singularities in the kernels. As a result, the integral equations are approximated by systems of linear algebraic equations with easy-to-compute coefficients, which are solved numerically by applying the generalized minimal residual method. A modification of the method is proposed that yields solutions in the spectra of interior Dirichlet problems and integral operators when the integral equations are not equivalent to the original differential problems and are not well-posed. Numerical results are presented for assessing the capabilities of the approach. 
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     author = {A. A. Kashirin and S. I. Smagin},
     title = {Potential-based numerical solution of {Dirichlet} problems for the {Helmholtz} equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
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     publisher = {mathdoc},
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     number = {8},
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     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_8_a10/}
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A. A. Kashirin; S. I. Smagin. Potential-based numerical solution of Dirichlet problems for the Helmholtz equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 8, pp. 1492-1505. http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_8_a10/