Legendre spectral method and error estimates for Helmholtz transmission eigenvalues in a cylinder

In this paper, an efficient Legendre spectral method based on a mixed formulation is proposed for Helmholtz transmission eigenvalues in a cylinder. To cope with the difficulty caused by the nonself-conjugacy of the problem, two auxiliary functions are first introduced to transform the initial problem into an equivalent fourth-order coupled form. Then the original three-dimensional problem is decomposed into a series of decoupled two-dimensional fourth-order transmission eigenvalue problems, with the help of the cylindrical coordinate transformation and the orthogonality of Fourier basis functions. By using the compact embedding theory of a class of weighted Sobolev spaces, a weak form and its discrete scheme for each two-dimensional fourth-order transmission eigenvalue problem are derived. Finally, error estimates for both the eigenvalues and eigenfunctions approximations are established, and the error is proved to converge exponentially with respect to the polynomial degree. Numerical experiments are provided to confirm the theoretical findings and to validate the efficiency of our algorithm.