A Class Of Spectral Method With The Tensor-product Nodal Basis For The Steklov Eigenvalue Problem

The spectral method is an important numerical method for solving differential equa-tions developed after finite-difference and finite element methods. The finite-differenceand finite element methods are local numerical methods, in practice, finite element meth-ods are particularly well suitable for problems in complex geometries, whereas the spectralmethod is a global method, which can provide a superior accuracy, at the expense of do-main flexibility. Because of characteristics of superior accuracy and low computation, itis widely used in the field of meteorology, physics, and mechanics (see[1,2]). However,to the best of our knowledge, there has been no reports on spectral methods for Stekloveigenvalue problems. In addtion, more and more scholars studied numerical method-s for Steklov eigenvalue problems in recent years. This paper first applies the spectralmethod to the Steklov eigenvalue problem, which discusses spectral method with thetensor-product nodal basis at the Legendre-Gauss-Lobatto points for solving the Stekloveigenvalue problem. A priori error estimates of the spectral method are discussed, andbased on the work of Melenk and Wohlmuth(Adv Comput Math15(2001), pp.311-331),an a posterior error estimator of the residual type is given and analyzed. In addition,inspired by the work of a two-grid discretization scheme based on shifted-inverse powermethod of Y. Yang and H. Bi([3–10]), this paper combines the shifted-inverse iterativemethod and the spectral method to establish an efficient scheme. Finally, numerical ex-periments with Matlab program are reported, and we also prepare the spectral methodand the bilinear finite element method from accuracy and verify the spectral method ishighly-efficient computational method.