Spectral Discretization of the Stokes Problem with Mixed Boundary Conditions

Abstract

The variational formulation of the Stokes problem with three independent unknowns, the vorticity, the velocity and the pressure, was born to handle non standard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity. We propose an extension of this formulation to the case of mixed boundary conditions in a three-dimensional domain. Next we consider a spectral discretization of this problem. A detailed numerical analysis leads to error estimates for the three unknowns and numerical experiments confirm the interest of the discretization. Résumé:La formulation variationnelle du problème de Stokes avec trois inconnues indépendantes, le tourbillon, la vitesse et la pression, est née pour traiter des conditions aux limites non usuelles portant sur la composante normale de la vitesse et les composantes tangentielles du tourbillon. Nous proposons une extension de cette formulation au cas de conditions aux limites mixtes dans un domaine tri-dimensionnel. Nous considérons ensuite une discrétisation obtenue par méthode spectrale. Une analyse numérique détaillée permet d'établir des majorations d'erreur pour les trois inconnues et des expériences numériques confirment l'intérêt de la discrétisation.

Least-squares formulations for Stokes equations with non-standard boundary conditions -- A unified approach

“…Convergence analysis and numerical simulations have been presented. A spectral discretization for vorticity based first order formulation of Stokes equations with (B4) has been discussed in [6].…”

Spectral discretization of the time‐dependent Stokes problem with mixed boundary conditions

“…We also assume that is a Lipschitz‐continuous submanifold of ∂ Ω. The basic idea in previous studies 1,2,5 consists of introducing the vorticity as a new unknown. Problem () can equivalently be written as (we suppress the variables and t for brevity), …”

“…We now intend to prove the well‐posedness of problems () and (), and we first introduce the kernel The proof of the next lemma is given in Amoura et al 5, Lem. 3.2 …”

“…We also assume that 𝜕Γ m = 𝜕Γ is a Lipschitz-continuous submanifold of 𝜕Ω. The basic idea in previous studies 1,2,5 consists of introducing the vorticity 𝝎 = curl u as a new unknown. Problem (1.1) can equivalently be written as (we suppress the variables x and t for brevity),…”

“…Spectral discretization of the stationary Stokes and Navier–Stokes equations with this type of conditions has been done in Amoura et al 5 and Daikh and Yakoubi 6 ; see also Bernardi and Sayah 4 for the finite element discretization. Aldbaissy et al 7 study the unsteady state coupled system of Navier–Stokes and heat equation by two types of discrete scheme: first and second order.…”