The Hermite-type virtual element method with interior penalty for the fourth-order elliptic problem

We present a Hermite-type virtual element method with interior penalty to solve the fourth-order elliptic problem over general polygonal meshes, where some interior penalty terms are added to impose the \(C^1\) continuity. A \(C^0\)-continuous Hermite-type virtual element with local \(H^2\) regularity is constructed, such that it can be used in the interior penalty scheme. We prove the boundedness of basis functions and interpolation error estimates of Hermite-type virtual element. After introducing a discrete energy norm, we present the optimal convergence of the interior penalty scheme. Compared with some existing methods, the proposed interior penalty method uses fewer degrees of freedom. Finally, we verify the theoretical results through some numerical examples.