An arbitrary-order discrete rot-rot complex on polygonal meshes with application to a quad-rot problem

TL;DR

This paper introduces a high-order discrete de Rham complex on polygonal meshes, enabling a new stable and convergent discretization method for the quad-rot problem without forcing term preparation.

Contribution

It develops a novel discrete de Rham complex with enhanced regularity supporting arbitrary polygonal meshes and orders, and applies it to a stable, convergent discretization of the quad-rot problem.

Findings

Exactness of the complex on contractible domains

Complete stability and convergence analysis

Numerical validation confirms theoretical results

Abstract

In this work, following the discrete de Rham (DDR) approach, we develop a discrete counterpart of a two-dimensional de Rham complex with enhanced regularity. The proposed construction supports general polygonal meshes and arbitrary approximation orders. We establish exactness on a contractible domain for both the versions of the complex with and without boundary conditions and, for the former, prove a complete set of Poincar\'e-type inequalities. The discrete complex is then used to derive a novel discretisation method for a quad-rot problem which, unlike other schemes in the literature, does not require the forcing term to be prepared. We carry out complete stability and convergence analyses for the proposed scheme and provide numerical validation of the results.