A High Accuracy Numerical Method and Error Analysis for Fourth Order Elliptic Eigenvalue Problems in Circular Domain

Abstract

In this paper, an efficient spectral method is applied to solve fourth order elliptic eigenvalue problems in circular domain. Firstly, we derive the essential pole conditions and the equivalent dimension reduction schemes of the original problem. Then according to the pole conditions, we define the corresponding weighted Sobolev spaces. Together with the minimax principle and approximation properties of orthogonal polynomials, the error estimates of approximate eigenvalues are proved. Thirdly, we construct an appropriate set of base functions contained in approximation spaces and establish the matrix formulations for the discrete variational form, whose mass matrix and stiff matrix are all sparse so that we can solve the numerical solutions efficiently. Finally, we provide some numerical experiments to validate the theoretical results and algorithms.