The interior penalty virtual element method for the biharmonic problem

Abstract

In this paper, an interior penalty virtual element method (IPVEM) is developed for solving the biharmonic problem on polygonal meshes. By modifying the existing H 2 H^2 -conforming virtual element, an H 1 H^1 -nonconforming virtual element is obtained with the same degrees of freedom as the usual H 1 H^1 -conforming virtual element, such that it locally has H 2 H^2 -regularity on each polygon in meshes. To enforce the C 1 C^1 continuity of the solution, an interior penalty formulation is adopted. Hence, this new numerical scheme can be regarded as a combination of the virtual element space and discontinuous Galerkin scheme. Compared with the existing methods, this approach has some advantages in reducing the degree of freedom and capability of handling hanging nodes. The well-posedness and optimal convergence of the IPVEM are proven in a mesh-dependent norm. We also derive a sharp estimate of the condition number of the linear system associated with IPVEM. Some numerical results are presented to verify the optimal convergence of the IPVEM and the sharp estimate of the condition number of the discrete problem. Besides, in the numerical test, the IPVEM has a good performance in computational accuracy by contrast with the other VEMs solving the biharmonic problem.