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

Abstract

We develop a novel Hybrid High-Order method for the incompressible Navier-Stokes problem robust for large irrotational body forces. The key ingredients of the method are discrete versions of the body force and convective terms in the momentum equation formulated in terms of a globally divergence-free velocity reconstruction. Denoting by λ the L 2 -norm of the irrotational part of the body force, the method is designed so as to mimick two key properties of the continuous problem at the discrete level, namely the invariance of the velocity with respect to λ and the nondissipativity of the convective term. The convergence analysis shows that, when polynomials of total degree ≤ k are used as discrete unknowns, the energy norm of the error converges as h k+1 (with h denoting, as usual, the meshsize), and the error estimate on the velocity is uniform in λ and independent of the pressure. The performance of the method is illustrated by a complete panel of numerical tests, including comparisons that highlight the benefits with respect to more standard HHO formulations.

A pressure‐robust weak Galerkin finite element method for Navier–Stokes equations

“…First we cite the following integration by parts formula and the estimates for projection operators, which will be used in the error estimate.Lemma Let $$$ D $$$ be a simply connected open polyhedral subset of $$$ \Omega $$$ . For all $$$ \mathbf{v},\mathbf{w},\mathbf{z}\in {\left[{H}^1(D)\right]}^d $$$ , it holds ( [46] , Proposition 1) $$$$ {\int}_D\left(\nabla \times \mathbf{w}\right)\times \mathbf{v}\cdot \mathbf{z}d\mathbf{x}={\int}_D\nabla \mathbf{w}\mathbf{v}\cdot \mathbf{z}d\mathbf{x}-{\int}_D\nabla \mathbf{w}\mathbf{z}\cdot \mathbf{v}. $$$$ Lemma For $$$ \mathbf{v}\in {W}^{s,r}(T) $$$ and all $$$ m\in \left\{0,\dots, s\right\} $$$, $$$$ {\left|\mathbf{v}-{\mathbf{Q}}_0\mathbf{v}\right|}_{W^{m,r}(T)}\lesssim {h}_T^{s-m}{\left|\mathbf{v}\right|}_{W^{s,r}(T)}.$$…”

“…First, by Helmholtz decomposition, we can denote $$$ \mathbf{f}=\mathbf{g}+\nabla \psi $$$, where $$$ \mathbf{g} $$$ is the curl of a function in $$$ \mathbf{H}\left(\operatorname{curl};\Omega \right) $$$ whose tangential trace vanishes on $$$ \mathrm{\partial \Omega } $$$ and $$$ \psi \in {H}^1\left(\Omega \right) $$$. As shown in [46], by taking $$$ \mathbf{v}=\mathbf{u} $$$, $$$ q=p-\psi $$$ in (4) and (5), one has $$$$ \nu {\left\Vert \nabla \mathbf{u}\right\Vert}^2=\left(\mathbf{g},\mathbf{u}\right)\le \left\Vert \mathbf{g}\right\Vert \left\Vert \mathbf{u}\right\Vert \le C\left\Vert \mathbf{g}\right\Vert \left\Vert \nabla \mathbf{u}\right\Vert, $$$$ where we have used Poincare theorem in the last step. Thus, the exact solution is only bounded by solenoidal part of $$$ \mathbf{f} $$$: $$$$ \left\Vert \nabla \mathbf{u}\right\Vert \lesssim {\nu}^{-1}\left\Vert \mathbf{g}\right\Vert .$$…”

“…Then (21) follows (10) and the above inequality. By the estimate [46, Proposition 3] and the norm‐equivalence (10), $$$$ {\left\Vert {\mathbf{R}}_{\mathbf{T}}\left(\mathbf{v}\right)\right\Vert}_{L^r{\left(\Omega \right)}^d}\lesssim \mid {\left\Vert \mathbf{v}|\right\Vert}_1\lesssim \mid \left\Vert \mathbf{v}\mid \right\Vert, $$$$ complete the proof.…”

“…The pressure‐robust scheme for NS equation has been proposed in the convective and rotational formulation for the time‐dependent NS equations solved by Taylor‐Hood, MINI, and Crouzeix–Raviart finite element. Then, velocity reconstruction operator have been utilized in the hybrid high‐order method for providing a irrotational force robust scheme [46]. The robustness is achieved by using high order gradient reconstruction in the rotation formulation, where $$$ {\mathrm{P}}^{2k+2}{(T)}^{d\times d} $$$ is needed to produce a non‐dissipative simulation.…”

“…The author proposed to project the velocity into the H(div) space and use this reconstruction only in the body force assembling for Stokes equations for fixing the classical finite element schemes. Late on, the reconstruction operator has been used in [1,7,16,[32][33][34]46] to solve incompressible fluid equations. The pressure-robust scheme for NS equation has been proposed in the convective and rotational formulation for the time-dependent NS equations solved by Taylor-Hood, MINI, and Crouzeix-Raviart finite element.…”