A posteriori estimates for approximations of time-dependent Stokes equations

Karakatsani, Fotini; Makridakis, Charalambos

doi:10.1093/imanum/drl036

Cited by 12 publications

(12 citation statements)

References 31 publications

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“…Stokes reconstruction. The Stokes reconstruction, introduced in [27], is a key point in the a posteriori error analysis. One of its main advantages in the error analysis is that this reconstruction restores the divergence free condition of the error quantity, which otherwise is lost, and thus allows the application of various analytical techniques to derive the estimates.…”

Section: Preliminariesmentioning

confidence: 99%

A posteriori error estimates for fully discrete schemes for the time dependent Stokes problem

Bänsch¹,

Karakatsani

Makridakis

2018

Calcolo

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This work is devoted to a posteriori error analysis of fully discrete finite element approximations to the time dependent Stokes system. The space discretization is based on popular stable spaces, including Crouzeix-Raviart and Taylor-Hood finite element methods. Implicit Euler is applied for the time discretization. The finite element spaces are allowed to change with time steps and the projection steps include alternatives that is hoped to cope with possible numerical artifices and the loss of the discrete incompressibility of the schemes. The final estimates are of optimal order in L ∞ (L 2) for the velocity error.

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Section: Preliminariesmentioning

confidence: 99%

A posteriori error estimates for fully discrete schemes for the time dependent Stokes problem

Bänsch¹,

Karakatsani

Makridakis

2018

Calcolo

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“…The estimators are based on the methodology developed in [2,3,28] and then further used in [5,11,12,22]. The key point is the definition of an auxiliary function, called reconstruction, i.e.…”

Section: Aimmentioning

confidence: 99%

A Posteriori Error Estimates for Pressure-Correction Schemes

Bänsch¹,

Brenner²

2016

SIAM J. Numer. Anal.

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In this thesis, a-posteriori error estimates for time discretizations of the incompressible time dependent Stokes equations by pressure-correction methods are presented. Pressure-correction methods are splitting schemes, which decouple the velocity and the pressure. As a result the Stokes equations reduce to much easier schemes, which can be solved by cost-efficient algorithm. In a first step, a-posteriori estimates for the instationary Stokes system, discretized by the two-step backward differential formula method (BDF2), are presented. This allows to compare the a-posteriori estimators of the discretized Stokes system with the estimators of the pressure-correction scheme. In the second part of the thesis rigorous proofs of global upper bounds for the incremental pressure correction scheme discretized by backward Euler scheme as well as for the two-step backward differential formula method (BDF2) in rotational form are presented. Moreover, rate optimality of the estimators are shown for velocity (in case of backward Euler and BDF2 in rotational form) and pressure (in case of Euler). Computational experiments confirm the theoretical results.

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“…It is interesting to note the successful application of the ideas of Section 2 in the case of time dependent Stokes equations, [18,19]. The problem with Stokes (and therefore with Navier-Stokes) is twofold.…”

Section: Remarks-extensionsmentioning

confidence: 99%