We develop a new multipoint stress mixed finite element method for linear elasticity with weakly enforced stress symmetry on simplicial grids. Motivated by the multipoint flux mixed finite element method for Darcy flow, the method utilizes the lowest order Brezzi-Douglas-Marini finite element spaces for the stress and the vertex quadrature rule in order to localize the interaction of degrees of freedom. This allows for local stress elimination around each vertex. We develop two variants of the method. The first uses a piecewise constant rotation and results in a cell-centered system for displacement and rotation. The second uses a piecewise linear rotation and a quadrature rule for the asymmetry bilinear form. This allows for further elimination of the rotation, resulting in a cell-centered system for the displacement only. Stability and error analysis is performed for both variants. First-order convergence is established for all variables in their natural norms. A duality argument is further employed to prove second order superconvergence of the displacement at the cell centers. Numerical results are presented in confirmation of the theory.

中文翻译：

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简单网格弹性的多点应力混合有限元方法

我们开发了一种新的多点应力混合有限元方法，用于在简单网格上具有弱强制应力对称性的线性弹性。受 Darcy 流的多点通量混合有限元方法的启发，该方法利用最低阶 Brezzi-Douglas-Marini 有限元空间作为应力和顶点正交规则，以定位自由度的相互作用。这允许在每个顶点周围消除局部应力。我们开发了该方法的两种变体。第一个使用分段恒定旋转并导致以单元为中心的位移和旋转系统。第二个使用分段线性旋转和不对称双线性形式的正交规则。这允许进一步消除旋转，导致仅用于位移的以单元为中心的系统。对这两种变体都进行了稳定性和误差分析。为所有变量在其自然范数中建立一阶收敛。进一步使用对偶论证来证明单元中心位移的二阶超收敛。给出了数值结果以证实该理论。