Novel $H^{\mathrm{dev}}(\mathrm{Curl})$-conforming elements on regular triangulations and Clough--Tocher splits for the planar relaxed micromorphic model

Abstract

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 $H^{\mathrm{dev}}(\mathrm{Curl},A)$ become necessary. We present two novel $H^{\mathrm{dev}}(\mathrm{Curl},A)$-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.