A novel spectral method and rigorous error analysis for the interior transmission problem

Abstract

The problem of internal transmission, which arises in inverse scattering theory, is a boundary value problem that holds significant importance in qualitative methods due to its practical applications. In this paper, we propose and analyze an efficient spectral method for solving the transmission problem. To begin with, the two-dimensional interior transmission problem is transformed into a sequence of one-dimensional interior transmission problems by employing polar coordinates and the orthogonality of Fourier series. Furthermore, by leveraging the Fredholm alternation theorem and the approximation properties of orthogonal projection operators in non-uniformly weighted Sobolev spaces, a rigorous error estimate for the approximate solution is achieved. Additionally, we also engaged in a detailed description regarding the efficient implementation of our algorithm. Finally, to validate both the theoretical results and the efficacy of the algorithm, we also provide several numerical examples.