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

TL;DR

This paper develops spectral-Galerkin methods for Stokes eigenvalue problems in polar geometries, providing optimal error estimates and extending the approach to elliptic domains, validated by numerical experiments.

Contribution

It introduces a novel spectral-Galerkin approach with weighted Sobolev spaces for polar geometries and derives optimal error bounds for both circular and elliptic domains.

Findings

Optimal error estimates for eigenvalues in polar geometries

Parallelizable reduction to one-dimensional problems

Numerical validation of theoretical results

Abstract

In this paper we propose and analyze spectral-Galerkin methods for the Stokes eigenvalue problem based on the stream function formulation in polar geometries. We first analyze the stream function} formulated fourth-order equation under the polar coordinates, then we derive the pole condition and reduce the problem on a circular disk to a sequence of equivalent one-dimensional eigenvalue problems that can be solved in parallel. The novelty of our approach lies in the construction} of suitably weighted Sobolev spaces according to the pole conditions, based on which, the optimal error estimate for approximated eigenvalue of each one dimensional problem can be obtained. Further, we extend our method to the non-separable Stokes eigenvalue problem in an elliptic domain and establish the optimal error bounds. Finally, we provide some numerical experiments to validate our theoretical results and algorithms.