One approach for the simulation of metamaterials is to extend an associated continuum theory concerning its kinematic equations, and the relaxed micromorphic continuum represents such a model. It incorporates the Curl of the nonsymmetric microdistortion in the free energy function. This suggests the existence of solutions not belonging to \( H ^1\), such that standard nodal \( H ^1\)-finite elements yield unsatisfactory convergence rates and might be incapable of finding the exact solution. Our approach is to use base functions stemming from both Hilbert spaces \( H ^1\) and \( H (\mathrm {curl})\), demonstrating the central role of such combinations for this class of problems. For simplicity, a reduced two-dimensional relaxed micromorphic continuum describing antiplane shear is introduced, preserving the main computational traits of the three-dimensional version. This model is then used for the formulation and a multi step investigation of a viable finite element solution, encompassing examinations of existence and uniqueness of both standard and mixed formulations and their respective convergence rates.

模拟超材料的一种方法是扩展有关其运动学方程的关联连续体理论，而松弛的微晶连续体代表这种模型。它在自由能函数中包含了非对称微扭曲的Curl。这表明存在不属于\（H ^ 1 \）的解，因此标准节点\（H ^ 1 \）-有限元的收敛速度不理想，可能无法找到精确的解。我们的方法是使用源自希尔伯特空间\（H ^ 1 \）和\（H（\ mathrm {curl}）\）的基函数，说明此类组合对于此类问题的核心作用。为简单起见，引入了描述反平面剪切的简化的二维松弛微晶连续体，保留了三维版本的主要计算特征。然后将该模型用于可行的有限元解决方案的制定和多步骤研究，包括检查标准和混合配方的存在性和唯一性以及它们各自的收敛速度。