An introduction to Hybrid High-Order methods with applications to incompressible fluid mechanics


Thursday, 7 March, 2019 - 15:00

When: Thursday March 7, 2019 at 3:00PM

Where: SISSA main Campus, Via Bonomea 265, Trieste, Room A-133

Speaker: Prof. Daniele Di Pietro, Univ. Montpellier, France

Title: An introduction to Hybrid High-Order methods with applications to incompressible fluid mechanics

Abstract: Originally introduced in [1], Hybrid High-Order (HHO) methods provide a framework for the discretisation of models based on PDEs with features that set it apart from traditional ones.

The construction hinges on discrete unknowns that are broken polynomials on the mesh and on its skeleton, from which two key ingredients are devised:
(i) Local reconstructions obtained by solving small, trivially parallel problems inside each element, and conceived so that their composition with the natural interpolator of sufficiently smooth functions yields a physics- and problem-dependent projector on local polynomial spaces;
(ii) Stabilisation terms that penalise residuals designed at the element level so as to ensure stability while preserving the approximation properties of the reconstruction.
These ingredients are combined to formulate local contributions, which are then assembled as in standard Finite Element methods.
From this construction, several appealing features ensue: the support of polytopal meshes and arbitrary approximation orders in any space dimension; an enhanced compliance with the physics; a reduced computational cost thanks to the compact stencil along with the possibility to locally eliminate a large portion of the unknowns.

In this presentation after establishing the setting for HHO methods on a model diffusion problem [2], we study their applications to the incompressible Navier--Stokes equations [3,4]. In this context, HHO methods have additional appealing features, namely: they satisfy an inf-sup stable condition on general meshes; they support both the weak and strong enforcement of velocity boundary conditions, improving the resolution of boundary layers; they are amenable to efficient computer implementations where a large subset of both velocity and pressure unknowns are eliminated by solving local problems inside each element; they satisfy local momentum and mass balances inside each element with equilibrated interface fluxes. Particular care is devoted to the design of the convective trilinear form, which mimics at the discrete level the non-dissipation property of the continuous one. The possibility to add a convective stabilisation term is also contemplated. Quasi-optimal error estimates are established under the usual small data assumption, and convergence by compactness arguments is proved when the latter do not hold. A thorough numerical validation is provided both to confirm the theoretical results and to assess the method in more physical configurations (including, in particular, the well-known two- and three-dimensional lid-driven cavity problem).