On $$[H^{1}]^{3 \times 3}$$, $$[H(\text {curl})]^3$$ and $$H(\text {sym Curl})$$ finite elements for matrix-valued Curl problems

Abstract In this work we test the numerical behaviour of matrix-valued fields approximated by finite element subspaces $$[{{\,\mathrm{\textit{H}^1}\,}}]^{3\times 3}$$ [ H 1 ] 3 × , $$[ H (\textrm{curl}{})]^3$$ ( curl ) and $$ (\textrm{sym}\, \textrm{Curl}{})$$ sym Curl for a linear abstract variational problem connected to relaxed micromorphic model. The formulation corresponding elements is introduced, followed benchmarks our conclusions. continuum model reduces continuity assumptions classical replacing full gradient microdistortion in free energy functional with Curl. This results larger solution space microdistortion, namely place $$[{{\,\mathrm{\textit{H}^1}\,}}]^{3 \times . conditions on can be further weakened taking only symmetric part As shown recent works, new appropriate then newly introduced gives rise differential complex theory.