Discrete tensor product BGG sequences: splines and finite elements

TL;DR

This paper introduces a systematic method for discretizing BGG complexes using tensor-product spline and finite element spaces on cubical meshes, enabling new constructions of complex differential operators.

Contribution

It provides a novel tensor-product discretization framework for BGG complexes, including Hessian, elasticity, and divdiv complexes, on arbitrary-dimensional cubical meshes.

Findings

Constructed Hessian, elasticity, and divdiv complexes using the proposed discretization.

Demonstrated the applicability of tensor-product spline and finite element spaces in complex discretizations.

Enabled systematic discretization of BGG complexes on arbitrary-dimensional cubical meshes.

Abstract

In this paper, we provide a systematic discretization of the Bernstein-Gelfand-Gelfand (BGG) diagrams and complexes over cubical meshes of arbitrary dimension via the use of tensor-product structures of one-dimensional piecewise-polynomial spaces, such as spline and finite element spaces. We demonstrate the construction of the Hessian, the elasticity, and the divdiv complexes as examples for our construction.