Efficient Spectral Method For Elliptic Eigenvalue Problem On Polar Geometry

In this paper,an efficient spectral method is used to solve the ellipse eigenvalue problem in the extreme geometric region.The polar region takes circle domains and annular regions as examples to solve the fourth-order and second-order elliptic eigenvalue problems.For the fourth-order ellipse eigenvalue problem in the 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.Elliptic eigenvalue problem for annular regions,a high precision numerical method based on Legendre-Galerkin spectral approximation is presented in a annular domain in this paper.One firstly transforms the original problem into a series of one-dimensional eigenvalue problems by using polar coordinate transformation.Then a set of appropriate basis functions are chosen such that the stiffness and mass matrices of the discrete variational formulation are all sparse.Then by using the spectral theory of compact operator,we give a rigorous error estimation of the approximate solutions.In addition,the numerical example is presented and the numerical results show that our algorithm is effective.