We consider the equilibrium equations for a linearized Cosserat material. We identify their structure in terms of a differential complex, which is isomorphic to six copies of the de Rham complex through an algebraic isomorphism. Moreover, 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 weaky 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 Cosserat material parameters. 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。通过一个微分复形识别其结构，该复形通过代数同构与六个拷贝的de Rham复形同构。此外，我们展示如何通过继承线性化弹性力学的结果来分析Cosserat材料。这两种视角分别导出了两类混合有限元方法，我们称之为强耦合方法和弱耦合方法。我们证明了两类方法的收敛性，特别关注改进的收敛率估计以及Cosserat材料参数趋近于零极限情形下的稳定性。数值验证表明，理论结果完全反映了方法的实际性能。