Finite element grad grad complexes and elasticity complexes on cuboid meshes

Abstract: This paper constructs two conforming finite element grad grad and elasticity complexes on the cuboid meshes. For the finite element grad grad complex, an $H<sup>2$ conforming finite element space, an $\boldsymbol{H}(\operatorname{curl}; \mathbb{S})$ conforming finite element space, an $\boldsymbol{H}(\operatorname{div}; \mathbb{T})$ conforming finite element space and an $\boldsymbol{L}<sup>2$ 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 $\boldsymbol{H}<sup>1$ conforming space and an $$\boldsymbol{H}(\operatorname{curl}\operatorname{curl}^{T}; \mathbb{S})$$ conforming space are constructed. Combining with an existing $$\boldsymbol{H}(\operatorname{div}; \mathbb{S}) \cap \boldsymbol{H}(\operatorname{div}\operatorname{div}; \mathbb{S})$$ element and $\boldsymbol{H}(\operatorname{div}; \mathbb{S})$ element, respectively, these finite element spaces form two different finite element elasticity complexes. The exactness of all the finite element complexes is proved.