The Hybrid High-Order Method for Polytopal Meshes

Daniele Antonio Di Pietro (1) , Jerome Droniou (2, 3)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 I3M - Institut de Mathématiques et de Modélisation de Montpellier
3 School of Mathematics

Hybrid High-Order (HHO) methods are new generation numerical methods for models based on Partial Differential Equations with features that set them apart from traditional ones. These include: the support of polytopal meshes including non star-shaped elements and hanging nodes; the possibility to have arbitrary approximation orders in any space dimension; an enhanced compliance with the physics; a reduced computational cost thanks to compact stencil and static condensation. This monograph provides an introduction to the design and analysis of HHO methods for diffusive problems on general meshes, along with a panel of applications to advanced models in computational mechanics. The first part of the monograph lays the foundation of the method considering linear scalar second-order models, including scalar diffusion, possibly heterogeneous and anisotropic, and diffusion-advection-reaction. The second part addresses applications to more complex models from the engineering sciences: non-linear Leray-Lions problems, elasticity and incompressible fluid flows.

Mots clés: Navier-Stokes; Leray-Lions; Advection-diffusion-reaction; Polytopal methods; Discrete functional analysis; Linear elasticity; Hybrid High-Order methods; Third Strang lemma; Virtual Element methods

Domaines: Analyse numérique [math.NA]

Daniele Antonio Di Pietro, Jerome Droniou. The Hybrid High-Order Method for Polytopal Meshes: Design, Analysis, and Applications. Springer International Publishing, In press, Modeling, Simulation and Applications series, 978-3-030-37202-6 (Hardcover) 978-3-030-37203-3 (eBook). ⟨10.1007/978-3-030-37203-3⟩. ⟨hal-02151813v2⟩