Virtual element method for semilinear hyperbolic problems on polygonal meshes

ABSTRACT

This article deals with the development of the virtual element method for the approximation of semilinear hyperbolic problems. For the space discretization, two different operators are used: the energy projection operator ${\mathrm{\Pi }}_{k,K}^{\mathrm{\nabla }}$ and an internal $L^2$-projection operator ${\mathrm{\Pi }}_{k,K}^0$. In order to deal with the time derivative, a Newmark scheme is employed; and the resulted fully discrete scheme is analysed. Moreover, with the help of projection operators, optimal error estimates are derived for both semi- and fully discrete schemes in $L^2$-norm and $H^1$-norm. We have conducted numerical experiments on polygonal meshes to illustrate the performance of the proposed scheme and validate the theoretical findings.

KEYWORDS:

• Virtual element method
• polygonal meshes
• Newmark scheme
• conforming methods
• error estimates
• numerical experiments

2010 AMS SUBJECT CLASSIFICATIONS:

• 65N30
• 65N12
• 65N20
• 65N22

Disclosure statement

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