In this paper we propose a novel way to prescribe weakly the symmetry of stress tensors in weak formulations amenable to the construction of mixed finite element schemes. The approach is first motivated in the context of solid mechanics (using, for illustrative purposes, the linear problem of linear elasticity), and then we apply this technique to reduce the computational cost of augmented fully-mixed methods for thermal convection problems in fluid mechanics, in the case where several additional variables are defined. We show that the new approach allows to maintain the same structure of the mathematical analysis as in the original formulations. Therefore we only need to focus on ellipticity of certain bilinear forms, as this property provides feasible ranges for the stabilization parameters that complete the description of augmented methods. In addition, we present some numerical examples to show that these methods perform better than their counterparts that include vorticity, and emphasize that the reduction in degrees of freedom (and therefore, in computational cost) does not affect the quality of numerical solutions.

中文翻译：

---

连续力学中混合有限元公式的应力超弱对称性

在本文中，我们提出了一种新颖的方法来在适合混合有限元格式的弱公式中弱地规定应力张量的对称性。该方法首先是在固体力学的背景下（出于说明目的，使用线性弹性的线性问题），然后应用此技术来减少流体力学中热对流问题的增强完全混合方法的计算成本，如果定义了几个其他变量。我们表明，新方法可以保持与原始公式相同的数学分析结构。因此，我们只需要关注某些双线性形式的椭圆度，因为此属性为完成扩充方法描述的稳定参数提供了可行的范围。