A New Div-Div-Conforming Symmetric Tensor Finite Element Space with Applications to the Biharmonic Equation

TL;DR

A new $H(\textrm{divdiv})-conforming finite element is presented, which avoids the need for super-smoothness by redistributing the degrees of freedom to edges and faces, which leads to a hybridizable mixed method with superconvergence for the biharmonic equation.

Abstract

A new $H(\textrm{divdiv})$-conforming finite element is presented, which avoids the need for super-smoothness by redistributing the degrees of freedom to edges and faces. This leads to a hybridizable mixed method with superconvergence for the biharmonic equation. Moreover, new finite element divdiv complexes are established. Finally, new weak Galerkin and $C^0$ discontinuous Galerkin methods for the biharmonic equation are derived.

Figures (2)

• Figure 1: Redistribution of vertex degrees of freedom to faces and edges. $\boldsymbol \tau (\texttt{v}_0)\in \mathbb S$ is a symmetric tensor containing $6$ components. Three diagonal entries $\boldsymbol n_{F_i}^{\intercal}\boldsymbol \tau (\texttt{v}_0)\boldsymbol n_{F_i}$ will be distributed to faces $F_i$ for $i=1,2,3$ and three off-diagonal entries $\boldsymbol n_{F_i}^{\intercal}\boldsymbol \tau (\texttt{v}_0)\boldsymbol n_{F_j}$ to the edges $e_{ij} = F_i\cap F_j$ with $1\leq i< j\leq 3$.
• Figure 2: The lowest degree pair $\Sigma_{1^{++}}(T;\mathbb S) - \mathbb P_1(T)$ in three dimensions.

Theorems & Definitions (70)

• Lemma 2.1: Lemma 5.2 in ChenHuangDivRn2022
• Lemma 2.2: Proposition 3.6 in Fuhrer;Heuer;Niemi:2019ultraweak
• Lemma 2.3
• proof
• Lemma 2.4: Theorem 5.10 in ChenHuangDivRn2022
• Remark 2.5
• Lemma 2.6
• proof
• Lemma 2.7
• proof