A posteriorierror analysis of the fully discretized time-dependent Stokes equations

Abstract

Abstract. The time-dependent Stokes equations in two-or three-dimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization.Mathematics Subject Classification. 65N30, 65N15, 65J15.

A comparison of duality and energy a posteriori estimates for $\mathrm {L}_{\infty }(0,T;\mathrm {L}_2(\varOmega ))$ in parabolic problems

“…This is not obvious, indeed, in many works aposteriori analysis, the elliptic part is entangled with the parabolic part and there is not a clear cut difference between elliptic and parabolic effects. As noted in recent work on aposteriori analysis for time-dependent problems (Akrivis et al, 2006;Bergam et al, 2005;Bernardi and Verfürth, 2004;de Frutos and Novo, 2002;Picasso, 1998, e.g. ) understanding the splitting between the elliptic, stationary, and parabolic, time-dependent, errors, as well as the part of the error where these effects are coupled, is important in designing adaptive methods and avoiding repetition.…”

A posteriori error estimates for the Large Eddy Simulation applied to incompressible fluids

“…To prove the upper bound, we follow the idea used by Bernardi and Verfurth [9] or Bernardi and Sayah [7] for the Stokes problem in order to uncouple time and space errors. But in this work, the non linear term coming from the Navier-Stokes system requires more sophisticated calculations.…”

“…As far as time-dependent models are concerned, a large number of contributions may be found. To cite only a few of them, we can refer, for example, to Ladevèze [20] for constitutive relation error estimators for time-dependent non-linear FE analysis, Verfürth [27] for the heat equation, Bernardi and Verfürth [9] for the time dependent Stokes equations, Bernardi and Süli [8] for the time and space adaptivity for the second-order wave equation, Bergam, Bernardi and Mghazli [4] for some parabolic equations, Ern and Vohralïk [15] for estimation based on potential and flux reconstruction for the heat equation, and Bernardi and Sayah [7] for the time dependent Stokes equations with mixed boundary conditions. In [22], Nassreddine and Sayah treated the time dependent Navier-Stokes equations in two dimensions.…”

“…To derive an a posteriori estimate between the solution u of problem (FV) and the solution ūh corresponding to the solutions ūn h of (F V n,h ), it suffices to get an a posteriori estimate between the solution u τ of Problem (P aux ) and ūh and to apply the triangle inequality using the previous theorem. In order to get an a posteriori error estimate between the solutions u τ and ūh , we introduce the operator Π (see [9] defined from X into itself as follows: For each v in X, Πv denotes the velocity w of the unique weak solution (w, r) in X × M of the Stokes problem: ∀t ∈ X, (∇w, ∇t) − (div t, r) = 0, ∀q ∈ M, (div w, q) = (div v, q).…”

An adaptive finite element method for a time‐dependent Stokes problem

“…In this article, we focus on an a posteriori error analysis of the time-dependent equations. Some works, such as Bernardi and Verfürth [7] and Bernardi and Sayah [8], have approached this topic using an Euler scheme in time and a conforming finite element in space. They derived two types of error We use bold letters to represent vectors or vector functions, and define L 2 (Υ) ≔ (L 2 (Υ)) 2 and H s (Υ) ≔ (H s (Υ)) 2 .…”