We consider the equilibrium equations for a linearized Cosserat material and provide two perspectives concerning well-posedness. First, the system can be viewed as the Hodge Laplace problem on a differential complex. On the other hand, we show how the Cosserat materials can be analyzed by inheriting results from linearized elasticity. Both perspectives give rise to mixed finite element methods, which we refer to as strongly and weakly coupled, respectively. We prove convergence of both classes of methods, with particular attention to improved convergence rate estimates, and stability in the limit of vanishing characteristic length of the micropolar structure. The theoretical results are fully reflected in the actual performance of the methods, as shown by the numerical verifications.

翻译：我们考虑线性化Cosserat材料的平衡方程，并从两个角度讨论其适定性问题。首先，该系统可视为微分复形上的Hodge Laplace问题。另一方面，我们展示了如何通过继承线性化弹性的结果来分析Cosserat材料。这两种视角分别导出了强耦合与弱耦合的混合有限元方法。我们证明了这两类方法的收敛性，特别关注改进的收敛速率估计，以及在微极结构特征长度趋于零时的稳定性极限。数值验证表明，理论结果完全体现在方法的实际性能中。