Finite elements for div and divdiv conforming symmetric tensors in arbitrary dimension

TL;DR

This paper develops new finite element methods for symmetric tensor fields that conform to div and divdiv operators in any dimension, enhancing the mathematical tools for complex tensor computations.

Contribution

It introduces novel div- and divdiv-conforming finite elements for symmetric tensors in arbitrary dimensions, with a new approach based on dual space characterization.

Findings

Constructed vector div-conforming finite elements.

Developed symmetric div-conforming finite elements.

Built divdiv conforming finite elements using dual spaces.

Abstract

Several div-conforming and divdiv-conforming finite elements for symmetric tensors on simplexes in arbitrary dimension are constructed in this work. The shape function space is first split as the trace space and the bubble space. The later is further decomposed into the null space of the differential operator and its orthogonal complement. Instead of characterization of these subspaces of the shape function space, characterization of the dual spaces are provided. Vector div-conforming finite elements are firstly constructed as an introductory example. Then new symmetric div-conforming finite elements are constructed. The dual subspaces are then used as build blocks to construct divdiv conforming finite elements.