A high accuracy spectral method based on min/max principle for biharmonic eigenvalue problems on a spherical domain

Abstract

In this study, we develop a high precision spectral method based on the min/max principle for biharmonic eigenvalue problems in the spherical domain. By analyzing the orthogonal spherical harmonic and approximation and using the min/max principle, we first deduce the error estimates of approximate eigenvalues. Then we construct an appropriate set of orthogonal spherical functions contained in H 2 0 (Ω) and establish the matrix formulations for the discrete variational form, whose mass matrix and stiff matrix are all sparse so that we can solve the numerical solutions efficiently. Finally, we provide some numerical experiments to validate that the theoretical results are correct.

Convergence analysis of an efficient spectral element method for Stokes eigenvalue problem

“…However, most of them are finite element methods, which are far from enough for high‐precision calculation. To overcome the problem, An and Luo used the polar coordinate transformation and variable separation techniques to construct a high‐precision spectral element algorithm of the biharmonic eigenvalue problem. Nevertheless, such method can not be applied to irregular areas.…”

A triangular spectral method for the Stokes eigenvalue problem by the stream function formulation

“…The Stokes eigenvalue problem was arised in the analysis of the stability of stationary solutions of the Navier‐Stokes equations in a convex polygon . In recent years, there are various numerical methods to solve the Stokes eigenvalue problem, such as mixed‐finite element methods on square domain , spectral method on a spherical domain.…”