An introduction to Hybrid High-Order methods

TL;DR

This chapter introduces Hybrid High-Order (HHO) methods, a new class of numerical techniques for PDEs that support high-order approximations on complex meshes, with proven convergence and efficient computation.

Contribution

It presents the construction, analysis, and applications of HHO methods, including convergence proofs and numerical examples, for various PDE problems.

Findings

Proven a priori convergence for HHO methods.

Effective discretization of nonlinear p-Laplace and diffusion-advection-reaction problems.

Numerical results demonstrating method accuracy and efficiency.

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.