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 posteriori estimates for the energy norm. We then consider two applications: the discretization of the nonlinear -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.
Cite
@article{arxiv.1703.05136,
title = {An introduction to Hybrid High-Order methods},
author = {Daniele A. Di Pietro and Roberta Tittarelli},
journal= {arXiv preprint arXiv:1703.05136},
year = {2017}
}