An introduction to Hybrid High-Order methods
Provides a comprehensive introduction to HHO methods for researchers in numerical PDEs, but is primarily tutorial in nature.
This chapter introduces Hybrid High-Order (HHO) methods for PDEs, supporting arbitrary approximation orders on polyhedral meshes and reduced computational cost. It proves a priori convergence and a posteriori error estimates for the Poisson equation, and demonstrates applications to nonlinear p-Laplace and diffusion-advection-reaction problems.
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.