The Hermite-type virtual element method for second order problem

In this paper, we develop the Hermite-type virtual element method to solve the second order problem. A Hermitetype virtual element of degree >= 3 is constructed, which can be taken as an extension of classical Hermite finite element to polygonal meshes. For this virtual element, we rigorously prove some inverse inequalities and the boundedness of basis functions. Further, we prove the interpolation error estimates. Based on a computable H 1- projection, we give the discrete formulation and prove the optimal convergence for the Hermite-type virtual element method. Finally, we show some numerical results to verify the convergence of Hermite-type virtual element. Additionally, compared with other virtual elements, both theoretical analysis and numerical experiments demonstrate that the Hermite-type virtual element has fewer global degrees of freedom and results in significant computational savings.