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An Introduction to Hybrid High-Order Methods

  • Daniele Antonio Di Pietro
  • Roberta Tittarelli
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 15)

Abstract

This chapter provides an introduction to Hybrid High-Order (HHO) methods. These are new generation numerical methods for PDEs with several advantageous features: the support of arbitrary approximation orders on general polyhedral meshes, the reproduction at the discrete level of relevant continuous properties, and a reduced computational cost thanks to static condensation and compact stencil. After establishing the discrete setting, we introduce the basics of HHO methods using as a model problem the Poisson equation. We describe in detail the construction, and prove a priori convergence results for various norms of the error as well as a posteriori estimates for the energy norm. We then consider two applications: the discretization of the nonlinear p-Laplace equation and of scalar diffusion-advection-reaction problems. The former application is used to introduce compactness analysis techniques to study the convergence to minimal regularity solution. The latter is used to introduce the discretization of first-order operators and the weak enforcement of boundary conditions. Numerical examples accompany the exposition.

Notes

Acknowledgements

This work was funded by Agence Nationale de la Recherche grant HHOMM (ref. ANR-15-CE40-0005-01).

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Daniele Antonio Di Pietro
    • 1
  • Roberta Tittarelli
    • 1
  1. 1.Institut Montpelliérain Alexander Grothendieck, CNRSUniversité MontpellierMontpellierFrance

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