This paper devises a novel lowest-order conforming virtual element method (VEM) for planar linear elasticity with the pure displacement/traction boundary condition. The main trick is to view a generic polygon $K$ as a new one $\widetilde{K}$ with additional vertices consisting of interior points on edges of $K$, so that the discrete admissible space is taken as the $V_1$ type virtual element space related to the partition $\{\widetilde{K}\}$ instead of $\{K\}$. The method is shown to be uniformly convergent with the optimal rates both in $H^1$ and $L^2$ norms with respect to the Lam\'{e} constant $\lambda$. Numerical tests are presented to illustrate the good performance of the proposed VEM and confirm the theoretical results.
翻译:本文设计了一种新的最低顺序的虚拟元素方法(VEM), 符合纯置换/ 引力边界条件的平面线性弹性。 主要的诀窍是将通用多边形$K$视为新的一个$\ loblytilde{K}$, 加上由位于K$边缘的内部点组成的额外顶点, 以便允许的离散空间被视作与分隔区有关的V_ 1美元类型虚拟元素空间, 而不是$@@@K$。 该方法与Lam\\\'{e} 常数$\lambda$的最佳汇率一致, 并用H$1$和$L%2$标值统一。 进行了数值测试, 以说明拟议的VEM的良好性能, 并证实了理论结果 。