An efficient spectral method for the Helmholtz transmission eigenvalues in polar geometries (Q2321130)

This paper is concerning the internal transmission eigenvalue problem of acoustic wave scattering in an inhomogeneous medium, contained in a 2D or 3D domain \(\Omega\), with eigenfunctions \(u,v\). It is interesting that \(u,\omega \in L^2(\Omega)\), but \(\omega-u \in H^2(\Omega)\). The material properties of the scattering objects are governed by the eigenvalues of the problem. High-precision calculation can be obtained by spectral methods. A fourth-order scheme in polar geometries is given in this paper. Some pole conditions and appropriate weighted Sobolev space are used to avoid polar singularities. A weak form of the problem is considered and periodic angular solutions are sought, by using Fourier expansions. The corresponding finite dimensional problems are obtained. Approximate eigenvalues are studied by obtaining specific error estimations. The main tools are some compactness results and the approximation theory of \textit{I. Babuška} and \textit{J. Osborn} [in: Finite element methods (Part 1). Amsterdam etc.: North-Holland. 641--787 (1991; Zbl 0875.65087)] on self-adjoint and positive-definite eigenvalue problems. A basic tool is also the Poincaré inequality. In the last part, some efficient algorithms are given, based on Legendre polynomials (used as special basis functions). Numerical and graphical results show the remarkable accuracy of the proposed method.