Solution of discontinuous interior Helmholtz problems by the boundary and shell element method

The Helmholtz equation governing an interior domain with shell discontinuities is not efficiently solvable by the traditional boundary element method. In this paper it is shown how the Helmholtz equation can be recast as an integral equation known as the boundary and shell integral equation. The application of collocation to the integral equation gives rise to a method termed the boundary and shell element method. The associated problem of finding the eigenvalues and eigenfunctions of the Helmholtz equation in a discontinuous domain via the same method is also considered. This leads to a non-linear eigenvalue problem. Such a problem may be solved through polynomial interpolation of the matrix components. In this paper methods for solving the Helmholtz equation and the associated eigenvalue problem are implemented and applied to a test problem