Efficient Numerical Methods For Second-order Elliptic Eigenvalue Problem In Polar Geometry

For the second-order elliptic eigenvalue problem in polar geometry,an effective finite element method and a finite difference method are proposed.For the finite element method,we derive the essential polar condition and its equivalent dimension reduction scheme by using polar coordinate transformation.Then,we introduce an appropriate weighted Sobolev space according to the polar condition,derive the weak form of the reduced dimension scheme and the corresponding discrete scheme,and prove the error estimates of approximation eigenvalues by combining with the spectral theory of the compact operators and the approximation properties of the interpolation operators.In addition,we construct a set of appropriate basis functions in the approximation space,and establish the corresponding matrix form of the discrete variational scheme,and its mass matrix and stiffness matrix are all sparse,so it can be solved effectively.Finally,we provide some numerical examples,and the numerical results show that our algorithm is very effective.Moreover,we also present a finite difference method based on reduced dimension scheme for the second-order elliptic eigenvalue problem with the singularly variable coefficients in the circular and annular domains.By using the polar transformation,the original problem is transformed into a series of equivalent one-dimensional eigenvalue problems.For each one-dimensional eigenvalue problem,by using the properties of Taylor expansion of the function,the polar condition and the boundary condition,we establish the corresponding difference scheme,and give the effective implementation of the algorithm.Numerical results show that our algorithm is also effective.