An Efficient Finite Element Method For Fourth Order Elliptic Eigenvalue Problems In Circular/Spherical Domains

In this paper,we propose an effective finite element method for the fourth order elliptic eigenvalue problems in circular/spherical domains.For the circular domain,firstly,we use polar coordinate transformation and the orthogonal property of Fourier basis functions,the original problem is transformed into a series of equivalent one-dimensional eigenvalue problems.Secondly,according to the expression of Laplacian in polar coordinate system,the polar condition is derived.According to the polar condition,an appropriate weighted Sobolev space is introduced.For each one-dimensional eigenvalue problem,the weak form and the corresponding discrete scheme are established.Finally,we prove the error estimates of approximation eigenvalues and eigenvectors by means of the spectral theory of compact operators and the approximation properties of piecewise cubic Hermite interpolation polynomials.For the spherical domain,we use spherical coordinate transformation and the orthogonal property of spherical harmonic function to transform the original problem into a series of equivalent one-dimensional eigenvalue problems.Then,according to the expression of Laplacian in spherical coordinate system,the polar condition is also derived.We introduce an appropriate weighted Sobolev space and establish the weak form of dimension reduction schemes and the corresponding discrete scheme.In addition,according to the basic properties of piecewise cubic Hermite interpolation polynomials,we construct a set of appropriate basis functions,so that the mass matrix and stiffness matrix in the discrete scheme are sparse.Finally,we provide some numerical examples to show the validity of the algorithm and the correctness of the theoretical results.