Novel Hᵈev(Curl)-conforming elements on regular triangulations and Clough--Tocher splits for the planar relaxed micromorphic model

In this work we present a consistent reduction of the relaxed micromorphic model to its corresponding two-dimensional planar model, such that its capacity to capture discontinuous dilatation fields is preserved. As a direct consequence of our approach, new conforming finite elements for Hᵈev(Curl,A) become necessary. We present two novel Hᵈev(Curl,A)-conforming finite element spaces, of which one is a macro element based on Clough--Tocher splits, as well as primal and mixed variational formulations of the planar relaxed micromorphic model. Finally, we demonstrate the effectiveness of our approach with two numerical examples.