A hybrid $$ H ^1\times H (\mathrm {curl})$$ finite element formulation for a relaxed micromorphic continuum model of antiplane shear

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.