This paper constructs two conforming finite element grad grad and elasticity complexes on the cuboid meshes. For the finite element grad grad complex, an $H^2$ 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 $\boldsymbol{ L}^2$ 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 $\boldsymbol{ H}^1$ 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.
Categories: math.NA cs.NA