Spectral-Galerkin approximation and optimal error estimate for biharmonic eigenvalue problems in circular/spherical/elliptical domains

Abstract

In this paper, we propose and analyze spectral-Galerkin methods for the biharmonic eigenvalue problem in circular/spherical/elliptical domains. We first analyze the eigenfunction formulated fourth-order equation under the polar coordinates, then we derive the pole condition and reduce the problem on a circular disk/sphere 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 biharmonic 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.