The interior penalty virtual element method for the fourth-order elliptic hemivariational inequality

Abstract

We develop an interior penalty virtual element method (IPVEM) for solving a Kirchhoff plate contact problem, which can be described by a fourth-order elliptic hemivariational inequality (HVI). The virtual element space in IPVEM is constructed by modifying the H2-conforming virtual element space, and the number of degrees of freedom is greatly reduced. To force the C1 continuity, the interior penalty scheme is adopted, that is, we introduce a penalty term for the jump of normal derivatives on mesh edges and two additional terms to guarantee the consistency and symmetry of the scheme. With certain assumptions, the well-posedness of the discrete problem is proved. Furthermore, a priori error estimation is established for the IPVEM for the fourth-order elliptic HVI, and we show that the lowest-order VEM achieves optimal convergence order. Finally, some numerical examples are presented to support the theoretical results.