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In this paper we propose and analyze spectral-Galerkin methods for the Stokeseigenvalue problem based on the stream function formulation in polargeometries. We first analyze the stream function} formulated fourth-orderequation under the polar coordinates, then we derive the pole condition andreduce the problem on a circular disk to a sequence of equivalentone-dimensional eigenvalue problems that can be solved in parallel. The noveltyof our approach lies in the construction} of suitably weighted Sobolev spacesaccording to the pole conditions, based on which, the optimal error estimatefor approximated eigenvalue of each one dimensional problem can be obtained.Further, we extend our method to the non-separable Stokes eigenvalue problem inan elliptic domain and establish the optimal error bounds. Finally, we providesome numerical experiments to validate our theoretical results and algorithms.

Spectral-Galerkin Approximation and Optimal Error Estimate for Stokes Eigenvalue Problems in Polar Geometries