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).

A Hybrid High-Order method for the incompressible Navier–Stokes problem robust for large irrotational body forces

“…In Section 4.4, we will leverage (6) with X successively equal to the mesh elements to derive a reformulation of the convective term that will inspire the design of a consistent and non-dissipative discrete trilinear form. The discrete counterpart of (7), expressed by (46) below, will play a key role both in deriving an a priori bound on the discrete velocity uniform in λ (see Lemma 7) and in proving the error estimate of Theorem 10. For the sake of completeness, before proceeding, we give a proof of Proposition 1.…”

“…where we have used the Hodge decomposition (4) of f followed by the velocity-invariance property (5) to conclude. Simplifying the terms involving the bilinear form b in the above expression, invoking the non-dissipativity property (7) to write t(u, u, u) = 0, and recalling the definition (2) of the bilinear form a, we can go on writing…”

“…An intermediate result of independent interest used in the analysis are novel Sobolev inequalities for the Raviart-Thomas-Nédélec velocity reconstruction. The proposed ideas apply to other hybrid methods for incompressible flows; see, e.g., [7,9,12,18,23,33,36] and references therein. Replicating similar properties with virtually divergence-free numerical methods [4,5] (see also [3,10]) may constitute an interesting path for future research.…”

“…Section 4 contains the formulation of the discrete problem and includes, in particular, the definition and properties of the novel discrete convective trilinear form. The convergence analysis is carried out in Section 5, while a complete panel of numerical tests is provided in Section 6, including a comparison with the standard HHO scheme of [7].…”

Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity

“…HHO formulations have been discussed in. 42,43 On the one hand, special emphasis has been devoted to the construction of pointwise divergence-free approximations in incompressible flows. 44,45 Recent results proposing a relaxed H(div)conforming discretization of the velocity field are available in.…”