A priori and a posteriori estimates for three‐dimensional Stokes equations with nonstandard boundary conditions

Abstract

A priori and a posteriori estimates for three-dimensional Stokes equations with non standard boundary conditions. 2010. <hal-00442468v2> A PRIORI AND A POSTERIORI ESTIMATES FOR THREE-DIMENSIONAL STOKES EQUATIONS WITH NON STANDARD BOUNDARY CONDITIONSHYAM ABBOUD † , FIDA EL CHAMI † AND TONI SAYAH ‡ Abstract. In this paper we study the Stokes problem with some non standard boundary conditions. The variational formulation decouples into a system velocity and a Poisson equation for the pressure. The continuous and corresponding discrete system do not need an inf-sup condition. Hence, the velocity is approximated with curl conforming finite elements and the pressure with standard continuous elements. Next, we establish optimal a priori and a posteriori estimates and we finish this paper with numerical tests.

Error estimates for three-dimensional Stokes problem with non-standard boundary conditions

“…Estimates (22), (26) and (28) are local, i.e., only involve the error in a neighborhood of κ or e. The indicators η κ , ξ κ and γ κ can be viewed as a measure for the error of the space discretization and can be used to adapt the mesh-size in space. For all details of the proofs, we refer to [1].…”

“…For an arbitrary κ ∈ τ h , we denote by η κ the diameter of κ and by ρ κ the diameter of the sphere inscribed in κ. Then η denotes the maximum of η κ and we assume that τ η is regular in the sense of Ciarlet [2]: there exists a constant σ independent of η such that sup κ∈τ η…”

“…Concernant les etimations d'erreur a posteriori, nous commençons par établir celles relatives à la pression par les méthodes usuelles. Celles de la vitesse se basent sur la décomposition suivante ; u − u h = ∇λ + w où λ ∈ H 1 0 (Ω) et w ∈ V 0 (pour la condition aux limites (2)). Ainsi, nous obtenons le système suivant vérifié uniquement par λ :…”

A posteriori error analysis of the time dependent Navier–Stokes equations with mixed boundary conditions

“…This kind of mixed boundary conditions are also considered in [1] and [11] for the studies of the a posteriori errors for the Stokes problem, and also in [4] for the stationary case.…”

Error studies of the Coupling Darcy-Stokes System with velocity-pressure formulation

“…For the coupling problem, we cite the works [2], [3], [8], [9], [16] and [21]. In [1] and [12], we treat the Stokes problem with non-standard boundary conditions and we introduce velocity-pressure weak formulation.…”