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From Finite Elements to Hybrid High-Order methods

Daniele A. Di Pietro, Jérôme Droniou

TL;DR

This work surveys polytopal methods with a focus on Hybrid High-Order (HHO) discretizations for Poisson-type problems. It starts from the nonconforming Crouzeix–Raviart scheme on conforming meshes and develops a general HHO framework that reconstructs potentials and gradients from face data on general polyhedral meshes, enabling arbitrary-order accuracy. The key contributions include the affine/polynomial reconstructions, the elliptic projector, and a stabilization strategy that yields a coercive, stable method with a rigorous error analysis showing optimal convergence for smooth solutions. The work demonstrates how polytopal ideas bypass mesh-conformity constraints, facilitate local refinement and coarsening, and provide a natural path to higher-order schemes on general meshes, with clear connections to classical CR methods in limiting cases.

Abstract

This document contains lecture notes from the Ph.D. course given at Scuola Superiore Meridionale by Daniele Di Pietro in February 2025. The goal of the course is to provide an overview of polytopal methods, focusing on the Hybrid High-Order (HHO) method. As a starting point, we study the Crouzeix-Raviart method for a pure diffusion equation, with particular focus on its stability. We then show that, switching to a fully discrete point of view, it is possible to generalize it first to polyhedral meshes and then to arbitrary order, leading to a method that belongs to the HHO family. A study of the stability and consistency of this method reveals the need for a stabilization term, for which we identify two key properties.

From Finite Elements to Hybrid High-Order methods

TL;DR

This work surveys polytopal methods with a focus on Hybrid High-Order (HHO) discretizations for Poisson-type problems. It starts from the nonconforming Crouzeix–Raviart scheme on conforming meshes and develops a general HHO framework that reconstructs potentials and gradients from face data on general polyhedral meshes, enabling arbitrary-order accuracy. The key contributions include the affine/polynomial reconstructions, the elliptic projector, and a stabilization strategy that yields a coercive, stable method with a rigorous error analysis showing optimal convergence for smooth solutions. The work demonstrates how polytopal ideas bypass mesh-conformity constraints, facilitate local refinement and coarsening, and provide a natural path to higher-order schemes on general meshes, with clear connections to classical CR methods in limiting cases.

Abstract

This document contains lecture notes from the Ph.D. course given at Scuola Superiore Meridionale by Daniele Di Pietro in February 2025. The goal of the course is to provide an overview of polytopal methods, focusing on the Hybrid High-Order (HHO) method. As a starting point, we study the Crouzeix-Raviart method for a pure diffusion equation, with particular focus on its stability. We then show that, switching to a fully discrete point of view, it is possible to generalize it first to polyhedral meshes and then to arbitrary order, leading to a method that belongs to the HHO family. A study of the stability and consistency of this method reveals the need for a stabilization term, for which we identify two key properties.

Paper Structure

This paper contains 26 sections, 12 theorems, 119 equations, 4 figures.

Table of Contents

  1. The Poisson problem
  2. On the stability of Galerkin approximations
  3. A non-conforming scheme for the Poisson problem on conforming simplicial meshes
  4. The Crouzeix--Raviart space
  5. Broken gradient
  6. A non-conforming scheme for the Poisson problem
  7. A magic formula
  8. The Raviart--Thomas--Nédélec space
  9. Proof of Lemma \ref{['lem:poincare:cr']}
  10. A hybrid high-order space on general meshes
  11. Limitations of the finite element approach
  12. Shifting point of view
  13. A lowest-order hybrid space on general meshes
  14. Affine potential reconstruction
  15. Interpolator and polynomial consistency of the affine potential reconstruction
  16. ...and 11 more sections

Key Result

Theorem 1

There exists a real number $C_\Omega > 0$ only depending on $\Omega$ such that, for all $v \in H_0^1(\Omega)$,

Figures (4)

  • Figure 1: Two successive local refinements of a conforming finite element mesh. The element to be refined at the next step is in gray. As the refinement proceeds, and unless specific measures are taken, more and more elongated elements, such as the one in red, appear.
  • Figure 2: Non-conforming local mesh refinement. If a method supporting polygonal meshes is available, the element in gray can simply be regarded as a pentagon, and it will not require a specific treatment.
  • Figure 3: Two examples of meshes with, respectively, 64 and 255 elements obtained by coarsening an initial $200 \times 200$ Cartesian orthogonal mesh.
  • Figure 4: Standard elements cut by an interface generate polygonal elements.

Theorems & Definitions (30)

  • Theorem 1: Continuous Poincaré inequality
  • Proposition 2: Stability of the Galerkin approximation
  • proof
  • Lemma 3: Discrete Poincaré inequality for the Crouzeix--Raviart space
  • Corollary 4: Stability of the Crouzeix--Raviart scheme
  • Proposition 5: Magic formula
  • proof
  • Remark 6: Validity on general meshes
  • Proposition 7: Divergence and normal trace of functions in $\mathcal{RT\!N}^1(T)$
  • proof
  • ...and 20 more