A Review of Hybrid High-Order Methods: Formulations, Computational Aspects, Comparison with Other Methods

Abstract

March 10, 2016Abstract Hybrid High-Order (HHO) methods are formulated in terms of discrete unknowns attached to mesh faces and cells (hence, the term hybrid), and these unknowns are polynomials of arbitrary order k ě 0 (hence, the term high-order). HHO methods are devised from local reconstruction operators and a local stabilization term. The discrete problem is assembled cellwise, and cell-based unknowns can be eliminated locally by static condensation. HHO methods support general meshes, are locally conservative, and allow for a robust treatment of physical parameters in various situations, e.g., heterogeneous/anisotropic diffusion, quasi-incompressible linear elasticity, and advectiondominated transport. This paper reviews HHO methods for a variable-diffusion model problem with nonhomogeneous, mixed Dirichlet-Neumann boundary conditions, including both primal and mixed formulations. Links with other discretization methods from the literature are discussed.

A primal discontinuous Galerkin method with static condensation on very general meshes

“…We shall show that the local problem (14) admits an infinity of solutions. We can choose any of these solutions on each mesh cell to form u loc h .…”

“…This is witnessed for example by the book [4]. The methods reviewed in this book (if we restrict our attention only to finite element type methods using piecewise polynomial approximation spaces in one form or another) include interior penalty discontinuous Galerkin (DG) methods [6,2], hybridizable discontinuous Galerkin (HDG) methods ( [8], introduced in [10]), the Virtual Element (VE) method ( [21], introduced in [20,5]), the Hybrid High-Order (HHO) method ( [14], introduced in [12,13]). One can add to this list the weak Galerkin finite element [22], which is similar to HDG.…”

A hybrid high‐order formulation for a Neumann problem on polytopal meshes

“…As low‐order methods on polyhedral meshes have been studied for quite some time, we mention that HHO can be seen as a finite volume method for polynomial of order k = 0 [1, section 2.5]. HHO can also be expressed into an equivalent mixed formulation .…”

“…As low-order methods on polyhedral meshes have been studied for quite some time, we mention that HHO can be seen as a finite volume method [3,4] for polynomial of order k = 0 [1, section 2.5]. HHO can also be expressed into an equivalent mixed formulation [5,6]. In [7], we find the identification of conservative numerical traces of the flux, and that HHO methods can be seen as a generalization of HDG methods.…”

“…In addition, thanks to the Poincaré-Wintinger inequality and the Lax-Milgram lemma, we can ensure that the problem (3) is well-posed. The application of HHO technique Poisson problem with Dirichlet conditions has been described in [9], while in [5] one can find the corresponding analysis for mixed boundary conditions. An extension of this approach for a certain class of nonlinear elliptic problems (p-Laplacian) has been developed in [10].…”