  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 H1, such that standard nodal H1-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 H1 and H(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.
