We introduce the harmonic virtual element method (VEM) (harmonic VEM), a modification of the VEM (Beirão da Veiga et al. (2013) Basic principles of virtual element methods. Math. Models Methods Appl. Sci., 23, 199–214.) for the approximation of the two-dimensional Laplace equation using polygonal meshes. The main difference between the harmonic VEM and the VEM is that in the former method only boundary degrees of freedom are employed. Such degrees of freedom suffice for the construction of a proper energy projector on (piecewise harmonic) polynomial spaces. The harmonic VEM can also be regarded as an ‘|$H^1$|-conformisation’ of the Trefftz discontinuous Galerkin-finite element method (TDG-FEM) (Hiptmair et al. (2014) Approximation by harmonic polynomials in starshaped domains and exponential convergence of Trefftz hp-DGFEM. ESAIM Math. Model. Numer. Anal., 48, 727–752.). We address the stabilization of the proposed method and develop an hp version of harmonic VEM for the Laplace equation on polygonal domains. As in TDG-FEM, the asymptotic convergence rate of harmonic VEM is exponential and reaches order |$\mathscr{O}(\exp (-b\sqrt [2]{N}))$|⁠, where |$N$| is the number of degrees of freedom. This result overperforms its counterparts in the framework of hp FEM (Schwab, C. (1998),p- and hp-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. Clarendon Press Oxford.) and hp VEM (Beirão da Veiga et al. (2018) Exponential convergence of the hp virtual element method with corner singularity. Numer. Math., 138, 581–613.), where the asymptotic rate of convergence is of order |$\mathscr{O}(\exp(-b\sqrt [3]{N}))$|⁠.

中文翻译：

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调和虚元法：多边形域上拉普拉斯问题的稳定性和指数收敛

我们引入谐波虚拟元件方法（VEM）（谐波VEM），所述VEM的变形例（Beirão达维加等人。（2013）的虚拟元件的方法的基本原理。数学模型的方法申请科学。，23，199- 214.），以便使用多边形网格逼近二维Laplace方程。谐波VEM和VEM之间的主要区别在于，在前一种方法中，仅采用边界自由度。这样的自由度足以在（分段谐波）多项式空间上构造适当的能量投影仪。谐波VEM也可以视为' | $ H ^ 1 $ | Trefftz不连续Galerkin有限元方法（TDG-FEM）的``符合化''（Hiptmair等。（2014）通过星形域中的谐波多项式逼近和Trefftz hp -DGFEM的指数收敛。ESAIM数学。模型。Numer。肛门 ，48，727-752。）。我们解决了所提出方法的稳定性问题，并为多边形域上的拉普拉斯方程开发了谐波VEM的hp版本。与TDG-FEM中一样，谐波VEM的渐近收敛速度是指数级的，达到| $ \ mathscr {O}（\ exp（-b \ sqrt [2] {N}））$ |⁠的阶，其中| $ N $ | 是自由度的数量。这一结果在hp FEM框架中的表现优于同类产品（Schwab，C.（1998），p和hp有限元方法：固体和流体力学中的理论和应用。Clarendon出版社牛津）和马力VEM（Beirão达维加等人。（2018）所述的指数收敛马力与角奇异虚拟元件的方法。NUMER。数学。，138，581-613。），其中收敛的渐近速率是顺序| $ \ mathscr {O}（\ exp（-b \ sqrt [3] {N}））$ |⁠。