A Spectral Galerkin Approximation for Second‐Order Non‐Selfadjoint Steklov Eigenvalue Problems in Complex Domains

ABSTRACT This article proposes a high‐order spectral‐Galerkin method for solving second‐order non‐selfadjoint Steklov eigenvalue problems on complex domains. A polar coordinate transformation is introduced to map the complex geometry onto a canonical disk, enabling the effective application of spectral methods to irregular domains. The equivalent transformed formulation of the original problem is derived, and the associated variational formulation and discrete scheme are developed. Rigorous a priori error estimates for the eigenvalues and eigenfunctions are presented. The discrete scheme is further expressed in matrix form to facilitate efficient numerical implementation. Numerical experiments demonstrate the spectral convergence and high‐order accuracy of the proposed method.