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An Introduction to Hybrid High-Order Methods

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Book cover Numerical Methods for PDEs

Part of the book series: SEMA SIMAI Springer 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.

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Acknowledgements

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

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Authors and Affiliations

  1. Institut Montpelliérain Alexander Grothendieck, CNRS, Université Montpellier, Montpellier, France

    Daniele Antonio Di Pietro & Roberta Tittarelli

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  1. Daniele Antonio Di Pietro
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  2. Roberta Tittarelli
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Correspondence to Daniele Antonio Di Pietro .

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Editors and Affiliations

  1. Institut Montpelliérain Alexander Grothendieck, CNRS, Université Montpellier, Montpellier, France

    Daniele Antonio Di Pietro

  2. Universit Paris Est, CERMICS (ENPC), Marne la Vallée, France

    Alexandre Ern

  3. MOX - Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

    Luca Formaggia

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Di Pietro, D.A., Tittarelli, R. (2018). An Introduction to Hybrid High-Order Methods. In: Di Pietro, D., Ern, A., Formaggia, L. (eds) Numerical Methods for PDEs. SEMA SIMAI Springer Series, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-319-94676-4_4

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