A review of Hybrid High-Order methods: formulations, computational aspects, comparison with other methods

Daniele Di Pietro (1) , Alexandre Ern (2) , Simon Lemaire (2)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

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>=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 advection-dominated 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.

Mots clés: High-order methods; General meshes; Variable-diffusion problems; Polytopal discretization methods

Domaines: Analyse numérique [math.NA]

Daniele Di Pietro, Alexandre Ern, Simon Lemaire. A review of Hybrid High-Order methods: formulations, computational aspects, comparison with other methods. G. R. Barrenechea, F. Brezzi, A. Cangiani, E. H. Georgoulis Eds. Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, Springer, 2016, Lecture Notes in Computational Science and Engineering. ⟨hal-01163569v3⟩