Hybrid High-Order Methods

Abstract

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C 0-hybrid high-order methods for biharmonic problems

“…This is straightforward for the right-hand side, whereas for the left-hand side, we use [22,Theorem 1.4.4.6] which gives lim δ→0 ∆ 2 v δ → ∆ 2 v in H −2+s (Ω). This proves the identity (7). □ Remark 2.2 (Literature) The identity from Lemma 2.1 is classical; see, e.g., [2].…”

“…These methods were introduced in [11] for linear diffusion and in [10] for locking-free linear elasticity. Since then, they have undergone a vigorous development, as reflected, e.g., in the two recent monographs [9,7]. Moreover, as discussed in [8,24,5], HHO methods are closely related to hybridizable discontinuous Galerkin (HDG) methods, weak Galerkin (WG) methods, nonconforming virtual element methods (ncVEM), and multiscale hybrid-mixed (MHM) methods.…”

A short perspective on a posteriori error control and adaptive discretizations

“…• recent polytopal methods such as Conforming Polygonal Finite Elements, see, e.g., Sukumar and Tabarraei (2004); Hybrid Discontinuous Galerkin (HDG), see, e.g., Cockburn et al (2016), Hybrid High Order (HHO) methods, see, e.g., Cicuttin et al (2021); Cockburn et al (2016); Di Pietro and Droniou (2020); Lemaire (2021), the Weak Galerkin Method, see, e.g., Dong and Ern (2022), the Virtual Element Method (VEM), see, e.g. Beirão da Veiga et al (2013); Lemaire (2021); the Smooth Finite Element Method (SFEM), see, e.g., Liu et al (2007); Nguyen-Xuan et al (2008b); modified discontinuous Galerkin with static condensation, see, e.g., Lozinski (2019);…”

Hybrid high-order methods for flow simulations in extremely large discrete fracture networks

“…Moreover, HHO methods have been bridged to hybridizable discontinuous Galerkin methods in [16]. HHO methods have been applied to a broad range of applications in computational mechanics and beyond; we refer the reader to the two recent textbooks [15,20] and the references therein. The main goal of the present work is to study the computational performance of HHO methods in the context of extremely large DFNs (over one million fractures).…”