Finite elements for divdiv-conforming symmetric tensors in three dimensions

TL;DR

This paper develops finite element spaces for divdiv conforming symmetric tensors in three dimensions, utilizing polynomial decompositions and trace characterizations to ensure conformity and continuity across elements.

Contribution

It introduces new finite element spaces for divdiv conforming symmetric tensors in 3D, including complexes and trace characterizations, advancing numerical methods for tensor equations.

Findings

Constructed divdiv conforming finite element spaces on tetrahedra.

Established divdiv and bubble function complexes in 3D.

Characterized traces for div-div operators and tensor continuity.

Abstract

Two types of finite element spaces on a tetrahedron are constructed for divdiv conforming symmetric tensors in three dimensions. The key tools of the construction are the decomposition of polynomial tensor spaces and the characterization of the trace operators. First, the divdiv Hilbert complex and its corresponding polynomial complexes are presented. Several decompositions of polynomial vector and tensors spaces are derived from the polynomial complexes. Then, traces for div-div operator are characterized through a Green's identity. Besides the normal-normal component, another trace involving combination of first order derivatives of the tensor is continuous across the face. Due to the smoothness of polynomials, the symmetric tensor element is also continuous at vertices, and on the plane orthogonal to each edge. Third, a finite element for sym curl-conforming trace-free tensors is constructed following the same approach. Finally, a finite element divdiv complex, as well as the bubble functions complex, in three dimensions are established.