This article presents a |$P_0$| finite element method for boundary value problems for linear elasticity equations. The new method makes use of piecewise constant approximating functions on the boundary of each polytopal element and is devised by simplifying and modifying the weak Galerkin finite element method based on |$P_1/P_0$| approximations for the displacement. This new scheme includes a tangential stability term on top of the simplified weak Galerkin to ensure the necessary stability due to the rigid motion. The new method involves a small number of unknowns on each element, it is user friendly in computer implementation and the element stiffness matrix can be easily computed for general polytopal elements. The numerical method is of second-order accurate, locking-free in the nearly incompressible limit, and ease polytopal partitions in practical computation. Error estimates in |$H^1$|⁠, |$L^2$| and some negative norms are established for the corresponding numerical displacement. Numerical results are reported for several two-dimensional and three-dimensional test problems, including the classical benchmark Cook’s membrane problem in two dimensions as well as some three-dimensional problems involving shear-loaded phenomena. The numerical results show clearly the simplicity, stability, accuracy and efficiency of the new method.

中文翻译：

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polytopal分区上线性弹性方程的一种无锁定P0有限元方法

本文介绍了一个|$P_0$| 线性弹性方程边值问题的有限元方法。新方法利用每个多面元边界上的分段常数逼近函数，并通过简化和修改基于|$P_1/P_0$|的弱伽辽金有限元方法而设计。位移的近似值。这个新方案包括在简化的弱伽辽金之上的切向稳定性项，以确保由于刚性运动而产生的必要稳定性。新方法涉及每个单元的少量未知数，它在计算机实现中对用户友好，并且可以轻松计算一般多面单元的单元刚度矩阵。该数值方法具有二阶精度，在几乎不可压缩的极限下无锁定，并且在实际计算中简化了多面划分。|$H^1$|⁠ , |$L^2$| 中的误差估计并为相应的数值位移建立了一些负范数。报告了几个二维和三维测试问题的数值结果，包括二维经典基准库克膜问题以及一些涉及剪切载荷现象的三维问题。数值结果清楚地表明了新方法的简单性、稳定性、准确性和效率。