Basic Principles of Hybrid High-Order Methods: The Poisson Problem

In this chapter we introduce the main ideas underlying HHO methods, using to this purpose the Poisson problem: Find \(u:\Omega \to \mathbb {R}\) such that $$\displaystyle \begin{aligned} \begin{array}{rcl} -{\Delta} u &= f \qquad \text{in}\ \Omega, \\ u &= 0 \qquad \text{on}\ \partial\Omega, \end{array} \end{aligned}$$ where Ω is an open bounded polytopal subset of \(\mathbb {R}^n\), n ≥ 2, with boundary ∂ Ω and \(f:\Omega \to \mathbb {R}\) is a given volumetric source term, assumed to be in L2( Ω).