Efficient Spectral Method And Aplication For The Eigenvalue Problem In Spherical Geometry

In this paper,we present the efficient spectral Galerkin approximation for circular and spherical domains,respectively.For the circular domain,we present spectral Galerkin approximation and rigorous error analysis for the Steklov eigenvalue problem.First of all,we use the polar coordinate transformation and technique of separation of variables to reduce the problem to a sequence of equivalent 1-dimensional eigenvalue problems that can be solved individually in parallel.Then,we derive the pole conditions and introduce weighted Sobolev space according to pole conditions.Together with the approximate properties of orthogonal polynomials,we prove the error estimates of approximate eigenvalues for each 1-dimensional eigen value problem.Finally,we provide some numerical experiments to validate the theoretical results and algorithms.For spherical domain,we present an efficient spectral method based on LegendreGalerkin approximation for the Helmholtz transmission eigenvalue problem.At first,By means of the spherical coordinate transformation and spherical harmonic expansion,we decompose the original problem into a sequence of equivalent one-dimensional generalized eigenvalue problems.Then we establish corresponding weak forms and discrete schemes for each one-dimensional generalized eigenvalue problem.Especially for the case of solid spherical domain,we need to deal with singularities introduced by spherical coordinate transformation.In order to overcome the difficulty,we derive the pole conditions and introduce weighted Sobolev space according to pole conditions.The corresponding weak forms and discrete schemes are also established according to the weighted Sobolev space.In addition,we prove the error estimates of eigenvalues and eigenfunctions by using the spectral theory of compact operators for each one dimensional generalized eigenvalue problems.Finally,we provide some numerical experiments to demonstrate the validity of the algorithm and the correctness of the theoretical results.