Bridging the hybrid high-order and virtual element methods

Abstract

We present a unifying viewpoint on hybrid high-order and virtual element methods on general polytopal meshes in dimension $2$ or $3$, in terms of both formulation and analysis. We focus on a model Poisson problem. To build our bridge (i) we transcribe the (conforming) virtual element method into the hybrid high-order framework and (ii) we prove $H^m$ approximation properties for the local polynomial projector in terms of which the local virtual element discrete bilinear form is defined. This allows us to perform a unified analysis of virtual element/hybrid high-order methods, that differs from standard virtual element analyses by the fact that the approximation properties of the underlying virtual space are not explicitly used. As a complement to our unified analysis we also study interpolation in local virtual spaces, shedding light on the differences between the conforming and nonconforming cases.

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);…”

C 0-hybrid high-order methods for biharmonic problems

“…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. Interestingly, the present C 0 -conforming HHO method (C 0 -HHO in short) can be viewed as a simple approach to extend C 0 -finite element methods for biharmonic problems by simply adding an additional unknown attached to the mesh faces representing the normal derivative of the solution.…”

A Hybrid High-Order Method for a Class of Strongly Nonlinear Elliptic Boundary Value Problems

“…HHO method in the lowest-order case falls in the family of the Hybrid Mixed Mimetic [40], which includes the Hybrid Finite Volume [44], the Mixed Finite Volume [37,38] and the Mixed-Hybrid Mimetic Finite Differences [17]. In [54], the author has bridged the HHO method with the virtual element method. We refer to [12,15,16,41,53] for related works.…”