We analyze a new stabilized dual-mixed method applied to incompressible linear elasticity problems, considering two kinds of data on the boundary of the domain: non homogeneous Dirichlet and mixed boundary conditions. In this approach, we circumvent the standard use of the rotation to impose weakly the symmetry of stress tensor. We prove that the new variational formulation and the corresponding Galerkin scheme are well-posed. We also provide the rate of convergence when each row of the stress is approximated by Raviart-Thomas elements and the displacement is approximated by continuous piecewise polynomials. Moreover, we derive a residual a posteriori error estimator for each situation. The corresponding analysis is quite different, depending on the type of boundary conditions. For known displacement on the whole boundary, we based our analysis on Ritz projection of the error, which requires a suitable quasi-Helmholtz decomposition of functions living in H(div;Ω). As a result, we obtain a simple a posteriori error estimator, which consists of five residual terms, and results to be reliable and locally efficient. On the other hand, when we consider mixed boundary conditions, these tools are not necessary. Then, we are able to develop an a posteriori error analysis, which provides us of an estimator consisting of three residual terms. In addition, we prove that in general this estimator is reliable, and when the traction datum is piecewise polynomial, locally efficient. In the second situation, we propose a numerical procedure to compute the numerical approximation, at a reasonable cost. Finally, we include several numerical experiments that illustrate the performance of the corresponding adaptive algorithm for each problem, and support its use in practice.

我们分析了一种新的稳定对偶混合方法应用于不可压缩线性弹性问题，考虑了域边界上的两种数据：非齐次 Dirichlet 和混合边界条件。在这种方法中，我们规避了旋转的标准使用来弱施加应力张量的对称性。我们证明了新的变分公式和相应的 Galerkin 方案是适定的。我们还提供了当每行应力由 Raviart-Thomas 元素近似并且位移由连续分段多项式近似时的收敛速度。此外，我们为每种情况推导出一个残差后验误差估计量。相应的分析是完全不同的，这取决于边界条件的类型。对于整个边界上的已知位移，H ( d i v;Ω)。结果，我们获得了一个简单的后验误差估计器，它由五个残差项组成，并且结果可靠且局部有效。另一方面，当我们考虑混合边界条件时，这些工具不是必需的。然后，我们能够开发一个后验误差分析，它为我们提供了一个由三个残差项组成的估计量。此外，我们证明了这个估计器通常是可靠的，并且当牵引数据是分段多项式时，局部有效。在第二种情况下，我们提出了一种以合理成本计算数值近似的数值程序。最后，我们包括几个数值实验，说明了每个问题的相应自适应算法的性能，并支持其在实践中的使用。