Finite element grad grad complexes and elasticity complexes on cuboid meshes

TL;DR

This paper develops new finite element complexes for grad grad and elasticity on cuboid meshes, proving their exactness and constructing conforming spaces with various regularities.

Contribution

It introduces novel conforming finite element spaces for grad grad and elasticity complexes on cuboid meshes, including reduced regularity versions, and proves their exactness.

Findings

Constructed $H^2$ conforming finite element space for grad grad complex.

Developed $oldsymbol{H}( ext{curl}; ext{S})$ and $oldsymbol{H}( ext{div}; ext{T})$ conforming spaces.

Proved the exactness of all constructed finite element complexes.

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 $H^2$ 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.