Oberwolfach report : Discretization of Hilbert complexes

Abstract

The report is based on an extended abstract for the MFO workshop "Hilbert Complexes: Analysis, Applications, and Discretizations", held at Oberwolfach during 19-25 June 2022. The aim is to provide an overview of some aspects of discretization of Hilbert complexes with an emphasis on confirming finite elements.Discretization of Hilbert complexes plays a central role in finite element exterior calculus [4,7,9]. In this report, we review some recent development in this direction, emphasizing conforming finite elements.

Discrete tensor product BGG sequences: splines and finite elements

“…For simplicity, we focus on the case n = 2. We discretize the 2D version of the second and third rows in the diagram (10) via two-dimensional spline spaces as follows: (42)…”

“…Recently, there has been a surge of interests on discretizing elasticity and other special cases of BGG complexes [2,20,21,22,19,38,39,40,47,25,27]. See also [42] for a review. Nevertheless, there seems to be no systematic discretization of either the BGG complexes or the entire BGG machinery behind them.…”

Nonlinear elasticity complex and a finite element diagram chase

“…The first column indicates the complexes that contain these spaces. See also [41] for a review of finite element Stokes complexes and [34] for some BGG complexes.…”