An efficient Legendre-Galerkin spectral approximation for the Steklov eigenvalue problem

In this paper, we give an effective spectral method based on the Legendre-Galerkin approximation for the Steklov eigenvalue problem. We first construct an appropriate set of basis functions such that the matrices in the discrete variational form are sparse. Then we deduce the matrix formulations based on the tensor-product for the discrete variational form in two- and three-dimensional cases, respectively. By using these tensor-product forms we can compute the discrete eigenvalues and eigenvectors rapidly. We also provide an error analysis and some numerical experiments. The numerical results indicate that our method is very stable and effective.