Finite Element Grad Grad Complexes and Elasticity Complexes on Cuboid Meshes

This paper constructs two conforming finite element grad grad and elasticity complexes on the cuboid meshes. For the finite element grad grad complex, an \(H^2\) $H^2$ conforming finite element space, an \(\varvec{H}({{\,\textrm{curl}\,}}; {\mathbb {S}})\) $\mathit{H}(\text{curl};\mathbb{S})$ conforming finite element space, an \(\varvec{H}({{\,\textrm{div}\,}}; {\mathbb {T}})\) $\mathit{H}(\text{div};\mathbb{T})$ conforming finite element space and an \(\varvec{L}^2\) ${\mathit{L}}^2$ finite element space are constructed. Further, a finite element complex with reduced regularity is also constructed, whose degrees of freedom for the three diagonal components are coupled. For the finite element elasticity complex, a vector-valued \(\varvec{H}^1\) ${\mathit{H}}^1$ conforming space and an \(\varvec{H}({{\,\textrm{curl}\,}}{{\,\textrm{curl}\,}}^{{\textsf{T}}}; {\mathbb {S}})\) $\mathit{H}(\text{curl}{\text{curl}}^{\mathsf{T}};\mathbb{S})$ conforming space are constructed. Combining with an existing \(\varvec{H}({{\,\textrm{div}\,}};{\mathbb {S}}) \cap \varvec{H}({{\,\textrm{div}\,}}{{\,\textrm{div}\,}};{\mathbb {S}})\) $\mathit{H}(\text{div};\mathbb{S})\cap \mathit{H}(\text{div}\text{div};\mathbb{S})$ element and an \(\varvec{H}({{\,\textrm{div}\,}}; {\mathbb {S}})\) $\mathit{H}(\text{div};\mathbb{S})$ element, respectively, these finite element spaces form two finite element elasticity complexes. The exactness of all the finite element complexes is proved.