A Hybrid High-Order method for the incompressible Navier--Stokes equations based on Temam's device

Abstract

In this work we propose a novel Hybrid High-Order method for the incompressible Navier--Stokes equations based on a formulation of the convective term including Temam's device for stability. The proposed method has several advantageous features: it supports arbitrary approximation orders on general meshes including polyhedral elements and non-matching interfaces; it is inf-sup stable; it is locally conservative; it supports both the weak and strong enforcement of velocity boundary conditions; it is amenable to efficient computer implementations where a large subset of the unknowns is eliminated by solving local problems inside each element. Particular care is devoted to the design of the convective trilinear form, which mimicks at the discrete level the non-dissipation property of the continuous one. The possibility to add a convective stabilisation term is also contemplated, and a formulation covering various classical options is discussed. The proposed method is theoretically analysed, and an energy error estimate in $h^{k+1}$ (with $h$ denoting the meshsize) is proved under the usual data smallness assumption. A thorough numerical validation on two and three-dimensional test cases is provided both to confirm the theoretical convergence rates and to assess the method in more physical configurations (including, in particular, the well-known two- and three-dimensional lid-driven cavity problems).