Setting

Abstract

In this chapter we introduce the setting for the development and analysis of Hybrid High-Order (HHO) methods. These methods are built upon general meshes possibly including polytopal elements and non-matching interfaces. In Sect. 1.1 we give a precise definition of polytopal mesh, and introduce the notion of regular sequence of h-refined polytopal meshes. In Sect. 1.2 we recall some basic notions on standard Lebesgue and Sobolev spaces, on the space H(div; Ω), and on polynomial spaces. We next introduce the first building block of HHO methods, namely local polynomial spaces, and prove some fundamental results for the analysis including, in particular, the comparison of Lebesgue and Sobolev (semi)norms defined on such spaces, as well as local trace inequalities valid on regular mesh sequences. Section 1.3 is devoted to the second key ingredient in HHO methods: projectors on local polynomial spaces.