| In this paper,we propose an efficient spectral-Galerkin method and a finite difference method based on dimensionality reduction scheme for the second-order elliptic eigenvalue problem in cylindrical geometry.For the spectral-Galerkin method,firstly,the second-order elliptic eigenvalue problem in rectangular coordinate system is transformed into an equivalent form in cylindrical coordinate system,and then the original problem is transformed into a series of two-dimensional eigenvalue problems in rectangular region by variable separation method.Secondly,for the two cases of solid cylinder and hollow cylinder,two Sobolev spaces and corresponding polynomial approximation spaces are introduced respectively,and the weak forms and discrete schemes are established for each reduced dimensional two-dimensional eigenvalue problems.Next,we prove the error estimation of the approximation eigenvalues using the spectral theory of fully continuous operators and the approximation properties of projection operators in non-uniform weighted Sobolev spaces.Furthermore,a set of valid basis functions in the approximation spaces is constructed and the matrix forms of the discrete schemes based on the tensor product were derived.At last,several numerical examples are given and the numerical results show that our algorithm is effective and high accurate.For the finite difference method,we propose a finite difference scheme based on dimensionality reduction scheme for the second-order elliptic eigenvalue problem with the singularly variable coefficients in the solid and hollow cylindrical geometries.First,a series of two-dimensional secondorder eigenvalue problems equivalent to the original problem are derived by using cylindrical coordinate transformation and generalized Fourier expansion.Then,by using the boundary conditions,the pole conditions and the Taylor expansion of the functions,a difference scheme for each twodimensional eigenvalue problem is established.Finally,we also give an effective implementation of the algorithm and some numerical examples,and numerical results show that our algorithm is effective. |
