In this paper, we develop the Hermite-type virtual element method to solve the second order problem. A Hermite-type 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 -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.

中文翻译：

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二阶问题的Hermite型虚元法


在本文中，我们开发了 Hermite 型虚拟单元方法来解决二阶问题。构造了≥3次的Hermite型虚拟单元，可以将其视为经典Hermite有限元对多边形网格的推广。对于这个虚拟元素，我们严格证明了一些逆不等式和基函数的有界性。此外，我们证明了插值误差估计。基于可计算的β投影，我们给出了Hermite型虚拟单元方法的离散公式并证明了其最优收敛性。最后，我们展示了一些数值结果来验证 Hermite 型虚拟单元的收敛性。此外，与其他虚拟单元相比，理论分析和数值实验都表明，Hermite型虚拟单元具有较少的全局自由度，并且可以显着节省计算量。