Spectral Galerkin Approximation and Error Analysis Based on a Mixed Scheme for Fourth-Order Problems in Complex Regions

In this article, an efficient spectral Galerkin method, which is based on a mixed scheme, is proposed and studied for solving fourth-order problems in complex regions. The fundamental idea behind this approach is to transform the initial problem into an equivalent form in cylindrical coordinates and to reshape the computational domain into a product-type rectangular one, which facilitates the utilization of spectral methods. However, when considering the equivalent fourth-order form directly in cylindrical coordinates, it introduces intricate pole conditions and variable coefficients, posing challenges to both theoretical analysis and algorithm implementation. To address this, we employ the orthogonality of Fourier series to further decompose it into a sequence of decoupled two-dimensional fourth-order eigenvalue problems. For each such problem, we introduce an auxiliary function to transform it into an equivalent second-order coupled system. Building on this, we formulate a mixed variational formulation and discrete scheme, and prove the error estimates for eigenvalue and eigenfunction approximations. Furthermore, we extend this algorithm to the two-dimensional complex domains. Finally, a series of numerical examples are presented, and the numerical results validate the effectiveness of the algorithm and the correctness of the theoretical results.