Spectral Method For Inhomogeneous Medium Acoustic Transmission Eigenvalue Problems And Steklov Eigenvalue Problem

The acoustic transmission eigenvalue problems are neither elliptical nor self conjugate,and there are certain difficulties in its theoretical analysis and numerical computation.For the Steklov eigenvalue problem in a spherical domain,the usual finite element method not only involves complex domain partitioning,but also introduces some errors in boundary approximation.Therefore,it is meaningful to propose a high-precision numerical method for the acoustic transmission eigenvalue problems and Steklov eigenvalue problem in a spherical domain.In this paper,we mainly focus on the spectral approximation of the acoustic transmission eigenvalue problems and Steklov eigenvalue problem.The content revolve around transmission eigenvalue problems in rectangle and cuboid,transmission eigenvalue problems in a spherical domain,Steklov eigenvalue problem in a spherical domain,respectively.We propose an efficient spectral method for the transmission eigenvalue problem in rectangle and cuboid.By rewriting the original problem into a equivalent fourth order coupled linear eigenvalue problem,we establish a variational formulation based on a mixed scheme.With the help of the spectral theory of the compact operator and the approximation property of the orthogonal projection operators in non-uniform weighted Soblev spaces,we prove the error estimates of the eigenvalues and eigenfunctions approximations.Numerical results show the convergence and spectral accuracy of the algorithm.We develop an efficient spectral-Galerkin approximation based on a dimensionality reduction for Helmholtz transmission eigenvalue problem in a spherical domain.By introducing an auxiliary function,we rewrite the original problem into an equivalent fourth order coupled form in spherical coordinates.Using the properties of spherical harmonic and Laplace-Beltrami operator,we further decompose the original problem into a series of one-dimensional fourth order coupled linear eigenvalue problems.Then we derive the essential polar conditions,define a class of weighted Sobolev spaces,and prove the compact embedding properties of each other for the first time.In addition,we establish a mixed variational formulation and its discrete variational formulation for each one-dimensional fourth order coupled linear eigenvalue problem.The error estimations of approximate eigenvalues and eigenfunctions are also proved by using the spectral theory of the compact operator.Finally,some numerical examples are presented to conform the theoretical error results and the efficiency of our algorithm.We present an efficient spectral method based on Legendre-Galerkin approximation for the Steklov eigenvalue problem in a spherical domain.Firstly,by means of spherical coordinate transformation and spherical harmonic expansion,the original problem is reduced to a sequence of equivalent decoupled one-dimensional eigenvalue problems.Through the introduction of the appropriate weighted Sobolev spaces,the weak form and corresponding discrete scheme are established for each one-dimensional eigenvalue problem.Then,we define some projection operators and prove their approximation properties,and then prove the error estimation of approximation eigenvalue.The numerical results show the effectiveness and spectral accuracy of our algorithm.