An introduction to Hybrid High-Order methods

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 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.
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Chapitre d'ouvrage
Daniele A. Di Pietro; Alexandre Ern; Luca Formaggia. Numerical Methods for PDEs – State-of-the-art Numerical Techniques, 15, Springer, 2018, SEMA-SIMAI, 978-3-319-94675-7 (Print) 978-3-319-94676-4 (eBook). 〈http://imag.edu.umontpellier.fr/event/ihp-nmpdes/〉
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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, 2018, SEMA-SIMAI, 978-3-319-94675-7 (Print) 978-3-319-94676-4 (eBook). 〈http://imag.edu.umontpellier.fr/event/ihp-nmpdes/〉. 〈hal-01490524〉

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