A C⁰ virtual element method for the biharmonic eigenvalue problem

Abstract

From the eigenvalue problem theory, we see that the convergence rate of the biharmonic eigenvalues obtained by the mixed method in I. Bab$\check{\mathrm{u}}$ska and J. Osborn, [Eigenvalue Problems, Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam, 1991.] is $h^{2k-2}$ for $k\ge 2$. In this paper, we give a presentation of the lowest-order virtual element method for the approximation of Kirchhoff plate vibration problem. This discrete scheme is based on a conforming $H^1\left(\mathrm{\Omega }\right)\times H^1\left(\mathrm{\Omega }\right)$ formulation, following the variational formulation of Ciarlet–Raviart method, which allows us to make use of simpler and lower-regularity virtual element space. By using the classical spectral approximation theory in functional analysis, we prove the spectral approximation and optimal convergence order $h^2$ for the eigenvalues. Finally, some numerical experiments are presented, which show that the proposed numerical scheme can achieve the optimal convergence order.

Keywords:

• Virtual element method
• biharmonic eigenvalue problem
• spectral approximation
• error analysis

2010 Mathematics Subject Classifications:

• 65N30
• 65N25
• 74K10

Acknowledgments

We wish to thank the referee for his/her constructive comments and suggestions. The work is supported by the Science Challenge Project (No.TZ2016002) and the Fundamental Research Funds for the Central Universities (No.xzy022019040).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work is supported by the Science Challenge Project (Grant Number TZ2016002) and the Fundamental Research Funds for the Central Universities (Grant Number xzy022019040).