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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.

An efficient finite-element method and error analysis for the fourth-order elliptic equation in a circular domain