Design and analysis of twisted and BGG Stokes-de Rham polytopal complexes

TL;DR

This paper introduces a novel discrete BGG diagram on polygonal meshes, combining Stokes and De Rham complexes within the DDR framework, and establishes their homological and analytical properties for applications in elasticity.

Contribution

It presents a new discrete BGG diagram on polygonal meshes, integrating Stokes and De Rham complexes, with proven homological and analytical properties for elasticity applications.

Findings

Established homological properties of the discrete Stokes complex.

Proved primal and adjoint consistency estimates and Poincaré inequalities.

Demonstrated applicability to generic polygonal meshes.

Abstract

We design a discrete Bernstein--Gelfand--Gelfand (BGG) diagram on polygonal meshes based on the DDR framework; the diagram is made of a discrete Stokes polygonal complex and a tensorised Discrete De Rham complex, and the BGG construction leads to a novel elasticity complex applicable on generic polygonal meshes. Complete homological and analytical properties of the discrete Stokes complex are established, including primal and adjoint consistency estimates as well as Poincar\'e inequalities. Homological properties of the complexes built from the BGG diagram are also established.