Novel $H^{\mathrm{d}\mathrm{e}\mathrm{v}}(\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l})$-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 $H^{\mathrm{d}\mathrm{e}\mathrm{v}}(\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l},A)$ become necessary. We present two novel $H^{\mathrm{d}\mathrm{e}\mathrm{v}}(\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l},A)$-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.