A serendipity fully discrete div-div complex on polygonal meshes

TL;DR

This paper introduces a reduced, fully discrete div-div complex on polygonal meshes using serendipity techniques, maintaining the original complex's properties while decreasing face degrees of freedom.

Contribution

It develops a novel serendipity-based reduction of a 2D elasticity complex, preserving its homological and analytical properties on polygonal meshes.

Findings

Reduced complex has same homological properties as original

New estimates for symmetric tensor-valued polynomial fields

Proved Poincaré--Korn-type inequalities for hybrid fields

Abstract

In this work we address the reduction of face degrees of freedom (DOFs) for discrete elasticity complexes. Specifically, using serendipity techniques, we develop a reduced version of a recently introduced two-dimensional complex arising from traces of the three-dimensional elasticity complex. The keystone of the reduction process is a new estimate of symmetric tensor-valued polynomial fields in terms of boundary values, completed with suitable projections of internal values for higher degrees. We prove an extensive set of new results for the original complex and show that the reduced complex has the same homological and analytical properties as the original one. This paper also contains an appendix with proofs of general Poincar\'e--Korn-type inequalities for hybrid fields.