A priori error analysis for finite element approximations of the Stokes problem on dynamic meshes

Brenner, Andreas; Bänsch, Eberhard; Bause, Markus

doi:10.1093/imanum/drt001

Cited by 15 publications

(7 citation statements)

References 17 publications

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“…In our case, this operator is the Lagrange interpolation operator which was also proposed in [40]. However, it is known that dynamically changing meshes may lead to spurious oscillations of the pressure for small time step sizes [16,18]. Indeed, we will observe those oscillations in our numerical results, as shown in Section 6.5.2.…”

Section: The Dirichlet-neumann Methodsmentioning

confidence: 73%

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Extended ALE Method for fluid–structure interaction problems with large structural displacements

Basting

Quaini

źanić

et al. 2017

Journal of Computational Physics

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Standard Arbitrary Lagrangian-Eulerian (ALE) methods for the simulation of fluid-structure interaction (FSI) problems fail due to excessive mesh deformations when the structural displacement is large. We propose a method that successfully deals with this problem, keeping the same mesh connectivity while enforcing mesh alignment with the structure. The proposed Extended ALE Method relies on a variational mesh optimization technique, where mesh alignment with the structure is achieved via a constraint. This gives rise to a constrained optimization problem for mesh optimization, which is solved whenever the mesh quality deteriorates. The performance of the proposed Extended ALE Method is demonstrated on a series of numerical examples involving 2D FSI problems with large displacements. Two way coupling between the fluid and structure is considered in all the examples. The FSI problems are solved using either a Dirichlet-Neumann algorithm, or a Robin-Neumann algorithm. The Dirichlet-Neumann algorithm is enhanced by an adaptive relaxation procedure based on Aitken's acceleration. We show that the proposed method has excellent performance in problems with large displacements, and that it agrees well with a standard ALE method in problems with mild displacement.

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Section: The Dirichlet-neumann Methodsmentioning

confidence: 73%

“…14 (a) occur whenever the ALE mapping is reparametrized. These jumps are to be expected for dynamically changing meshes, as pointed out in [16,18]. Next, we quantify the unbalance in the power exchange at the interface.…”

Section: Power Exchange At the Interfacementioning

confidence: 77%

Extended ALE Method for fluid–structure interaction problems with large structural displacements

Basting

Quaini

źanić

et al. 2017

Journal of Computational Physics

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“…The error estimates are presented first for static space meshes. For time-varying meshes, there is an additional projection error due to mesh changes, for which we derive a novel sharp estimate compared to existing results [43,4,20,10]. Our second main result are estimates on the error derivatives; we focus on static space meshes for simplicity.…”

Section: Introductionmentioning

confidence: 94%

Discontinuous Galerkin method in time combined with a stabilized finite element method in space for linear first-order PDEs

Ern

Schieweck²

2016

Math. Comp.

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We analyze the discontinuous Galerkin method in time combined with a finite element method with symmetric stabilization in space to approximate evolution problems with a linear, first-order differential operator. A unified analysis is presented for space discretization, including the discontinuous Galerkin method and H 1-conforming finite elements with interior penalty on gradient jumps. Our main results are error estimates in various norms for smooth solutions. Two key ingredients are the post-processing of the fully discrete solution by lifting its jumps in time and a new time-interpolate of the exact solution. We first analyze the L ∞ (L 2) (at discrete time nodes) and L 2 (L 2) errors and derive a super-convergent bound of order (τ k+2 + h r+1/2) for static meshes for k ≥ 1. Here, τ is the time step, k the polynomial order in time, h the size of the space mesh, and r the polynomial order in space. For the case of dynamically changing meshes, we derive a novel bound on the resulting projection error. Finally, we prove new optimal bounds on static meshes for the error in the time-derivative and in the discrete graph norm.

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“…The error introduced by this transfer operator will be accounted for by the error estimators anyway. However, due to the high sensitivity of the pressure approximation to mesh modification, see for instance [9,12,13], in order to avoid severe pressure oscillations, it is highly recommended to project the approximations on a given time level to the discretely divergence free space of the next time. To this end we introduce the following projection.…”

Section: Preliminariesmentioning

confidence: 99%

“…Changing the mesh (usually) results in a change of the discrete constraint, which in turn may have a devastating effect at least on the Lagrangian multiplier (i.e. the pressure for incompressible fluids), if the transfer of the primal variable from one step to the next one is realized by just standard operators (interpolation or L 2 projection), see [9,12,13].…”

Section: Introductionmentioning

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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A priori error analysis for finite element approximations of the Stokes problem on dynamic meshes

Cited by 15 publications

References 17 publications

Extended ALE Method for fluid–structure interaction problems with large structural displacements

Extended ALE Method for fluid–structure interaction problems with large structural displacements

Discontinuous Galerkin method in time combined with a stabilized finite element method in space for linear first-order PDEs

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

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