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In this paper, we propose an effective finite element method for the fourth order elliptic eigenvalue problems in a circular domain. First, by using polar coordinates transformation and the orthogonal property of Fourier basis functions, the original problem is turned into a series of equivalent one-dimensional eigenvalue problems. Second, according to the properties of Laplace operator in polar coordinate, we deduce the polar conditions and introduce suitable weighted Sobolev space. Based on the polar conditions and weighted Sobolev space, we establish the weak form and the corresponding discrete form. Third, we prove the error estimates of approximation eigenvalues and eigenvectors by means of the spectral theory of compact operators for each one-dimensional eigenvalue system. Finally, we provide some numerical experiments to show the validity of the algorithm and the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose an efficient finite-element method based on dimension reduction scheme to solve the fourth-order elliptic equation in a circular domain. First of all, by using polar coordinate transformation and Fourier basis function expansion, the necessary pole conditions are derived and the original problem is decomposed into a series of equivalent one-dimensional fourth-order problems. Based on the pole conditions, the appropriate weighted Sobolev spaces are introduced and the error estimate of finite-element solution is proved for each one-dimensional fourth-order problem. Then we construct the cubic Hermite interpolation basis functions and derive the matrix form corresponding to the discrete scheme. Furthermore, the sparsity of mass matrix and stiffness matrix are illustrated. Finally, we provide some numerical experiments, and the numerical results show that our method is very effective. [ABSTRACT FROM AUTHOR]