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 An efficient numerical method based on Legendre-Galerkin approximation for the Steklov eigenvalue problem in spherical domain Ting Tan,Jing An Keywords:Steklov eigenvalue problem, Legendre-Galerkin approximation, weighted Soblev space, error estimation, spherical domain Abstract: We present in this paper an efficient numerical method based on Legendre-Galerkin approximation for the Steklov eigenvalue problem in spherical domain. Firstly, by means of spherical coordinate transformation and spherical harmonic expansion, the original problem is reduced to a sequence of equivalent one-dimensional eigenvalue problems that can be solved individually in parallel. Through the introduction of the appropriate weighted Sobolev spaces, the weak form and corresponding discrete scheme are established for each one-dimensional eigenvalue problems. Then from the approximate property of orthogonal polynomials in the weighted Sobolev space, we prove the error estimates of approximate eigenvalues for each one dimensional eigenvalue problem. Finally, some numerical examples are provided to illustrate the validity of our algorithms.