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

Daniele A. Di Pietro, Marien-Lorenzo Hanot

TL;DR

This work develops a discrete divdiv complex in three dimensions on general polyhedral meshes using the DDR framework. By building entity-attached polynomial spaces and integration-by-parts-inspired operators, the authors establish local exactness on topologyically simple elements and design a stable mixed discretization scheme for the biharmonic problem with proven stability and an $h^{k+1}$ convergence rate. The contributions include new direct decompositions for matrix-valued face spaces, trimmed polynomial spaces, and a global Hessian reconstruction with a polynomially consistent $L^2$-like product, enabling a robust discretization on polyhedral meshes. The results offer flexible meshing capabilities and rigorous guarantees for the numerical treatment of high-order PDEs such as the biharmonic problem, with demonstrated numerical convergence on multiple mesh families.

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.

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

TL;DR

This work develops a discrete divdiv complex in three dimensions on general polyhedral meshes using the DDR framework. By building entity-attached polynomial spaces and integration-by-parts-inspired operators, the authors establish local exactness on topologyically simple elements and design a stable mixed discretization scheme for the biharmonic problem with proven stability and an convergence rate. The contributions include new direct decompositions for matrix-valued face spaces, trimmed polynomial spaces, and a global Hessian reconstruction with a polynomially consistent -like product, enabling a robust discretization on polyhedral meshes. The results offer flexible meshing capabilities and rigorous guarantees for the numerical treatment of high-order PDEs such as the biharmonic problem, with demonstrated numerical convergence on multiple mesh families.

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.
Paper Structure (45 sections, 12 theorems, 179 equations, 3 figures)

This paper contains 45 sections, 12 theorems, 179 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. Mesh
  4. Local and broken polynomial spaces
  5. Direct decompositions of local polynomial spaces
  6. Trimmed local polynomial spaces
  7. Reconstruction of tangent derivatives on edges
  8. Notation and basic results on traces
  9. Integration by parts formulas
  10. Integration by parts formulas for $\boldsymbol{H}^1(T;\mathbb{R}^3)$
  11. Integration by parts formulas for $\boldsymbol{H}(\mathop{\mathrm{\bf sym}}\nolimits\mathop{\mathrm{\bf curl}}\nolimits,T;\mathbb{T})$
  12. Integration by parts formulas for $\boldsymbol{H}(\mathop{\mathrm{div}}\nolimits\mathop{\mathrm{\bf div}}\nolimits,T;\mathbb{S})$
  13. Discrete divdiv complex
  14. Local discrete spaces
  15. Local operators
  16. ...and 30 more sections

Key Result

Lemma 1

For all $F\in\mathcal{F}_h$ and all $\ell\ge 0$, the following direct decomposition holds: where $\mathrm{adj}$ is the adjugate operator acting on $2\times 2$ matrices.

Figures (3)

  • Figure 1: Cubic mesh family
  • Figure 2: Tetgen tetrahedral mesh family
  • Figure 3: Voronoi mesh family

Theorems & Definitions (31)

  • Lemma 1: Direct decomposition of matrix-valued polynomial fields on faces
  • proof
  • Lemma 2: Link between $\boldsymbol{\mathcal{SR}}^{\ell}(T)$ and $\boldsymbol{\mathcal{H}}^{{\rm c},\ell+1}(T)$
  • proof
  • Remark 3: Extension to negative indices
  • Lemma 4: Traces of trimmed spaces
  • Lemma 5: Local commutation property
  • proof
  • Theorem 6: Local complex property and exactness
  • Remark 7: Role of the condition $k\ge 1$
  • ...and 21 more