A discrete three-dimensional divdiv complex on polyhedral meshes with application to a mixed formulation of the biharmonic problem

TL;DR

This paper develops a high-order discrete divdiv complex on polyhedral meshes using the DDR paradigm, enabling stable and convergent mixed formulations for biharmonic problems.

Contribution

It introduces a novel discrete divdiv complex on polyhedral meshes based on polynomial spaces and mimicking integration by parts, with applications to biharmonic problem discretization.

Findings

The complex is exact on trivial topology meshes.

The scheme is stable and convergent for biharmonic problems.

Provides detailed algebraic and numerical analysis.

Abstract

In this work, following the Discrete de Rham (DDR) paradigm, we develop an arbitrary-order discrete divdiv complex on general polyhedral meshes. The construction rests 1) on discrete spaces that are spanned by vectors of polynomials whose components are attached to mesh entities and 2) on discrete operators obtained mimicking integration by parts formulas. We provide an in-depth study of the algebraic properties of the local complex, showing that it is exact on mesh elements with trivial topology. The new DDR complex is used to design a numerical scheme for the approximation of biharmonic problems, for which we provide detailed stability and convergence analyses.