Spectral approximation and error analysis for the transmission eigenvalue problem with an isotropic inhomogeneous medium (Q6591547)

The article deals with the numerical solution of the transmission eigenvalue problem with a circular domain in an isotropic inhomogeneous medium. The problem is first transformed into polar coordinates, followed by the derivation of special polar conditions and the formulation of a mixed variational problem using a class of non-uniformly weighted Sobolev spaces. To solve the variational problem, the authors introduce a high-order Legendre-Fourier spectral method. Based on the spectral theory of compact operators, they establish abstract spectral approximation results and error estimates for both the eigenvalues and eigenvectors approximation. Furthermore, the paper provides a detailed description of implementing the resulting discrete scheme. Numerical experiments are performed on transmission problems in homogeneous, radial inhomogeneous, and general inhomogeneous media to illustrate the convergence and applicability of the method.