An introduction to Hybrid High-Order methods

Daniele Antonio Di Pietro (1) , Roberta Tittarelli (1)

1 IMAG - Institut Montpelliérain Alexander Grothendieck

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 posteri-ori estimates for the energy norm. We then consider two applications: the discretiza-tion 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.

Mots clés: p-Laplace equation; diffusion-advection-reaction; Poisson equation; Hybrid High-Order methods

Domaines: Analyse numérique [math.NA]

Daniele Antonio Di Pietro, Roberta Tittarelli. An introduction to Hybrid High-Order methods. Daniele A. Di Pietro; Alexandre Ern; Luca Formaggia. Numerical Methods for PDEs: State-of-the-art Numerical Techniques, 15, Springer, pp.75-128, 2018, SEMA SIMAI Springer Series, 978-3-319-94675-7 (Print) 978-3-319-94676-4 (eBook). ⟨10.1007/978-3-319-94676-4_4⟩. ⟨hal-01490524⟩