An unfitted Eulerian finite element method for the time-dependent Stokes problem on moving domains

Wahl, Henry von; Richter, Thomas; Lehrenfeld, Christoph

doi:10.1093/imanum/drab044

Cited by 19 publications

(7 citation statements)

References 43 publications

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“…for all appropriately smooth 𝑞 with ∇𝑞 ∈ 𝐷(0, 𝑇 ). By the definition of weak derivative (56) we have that…”

Section: Strong and Weak Materials Derivativementioning

confidence: 99%

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An evolving space framework for Oseen equations on a moving domain

Djurdjevac,

Gräser,

Herbert

2023

ESAIM: M2AN

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<p>This article considers non-stationary incompressible linear fluid equations in a moving domain. We demonstrate the existence and uniqueness of an appropriate weak formulation of the problem by making use of the theory of time-dependent Bochner spaces. It is not possible to directly apply established evolving Hilbert space theory due to the incompressibility constraint. After we have established the well-posedness, we derive and analyse a first order time discretisation of the system.</p>

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“…for all appropriately smooth 𝑞 with ∇𝑞 ∈ 𝐷(0, 𝑇 ). By the definition of weak derivative (56) we have that…”

Section: Strong and Weak Materials Derivativementioning

confidence: 99%

“…where we have used that 𝜑 * 𝑡 𝜂 := 𝐷Φ 𝑇 𝑡 𝜂 ∘ Φ 𝑡 . Therefore, by recalling the definition of a weak derivative (56), one may conclude that for a.e 𝑡, as an element of 𝑉 * (𝑡),…”

Section: Solution Spacementioning

confidence: 99%

An evolving space framework for Oseen equations on a moving domain

Djurdjevac,

Gräser,

Herbert

2023

ESAIM: M2AN

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“…They have been analyzed and studied numerically for parabolic problems, using low order finite elements and BDF time stepping schemes; compare for example, Reference 1. This analysis was recently extended to the time-dependent Stokes problem using, for the spatial discretization, lowest order Taylor-Hood elements 2 or equal-order elements along with stabilization, 3 combined with BDF time-stepping schemes. For instance, we refer to Reference 2 (Thm.…”

Section: Introductionmentioning

confidence: 99%

Cut finite element methods and ghost stabilization techniques for space‐time discretizations of the Navier–Stokes equations

Anselmann

Bause

2022

Numerical Methods in Fluids

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We propose and analyze numerically a new fictitious domain method, based on higher order space-time finite element discretizations, for the simulation of the nonstationary, incompressible Navier-Stokes equations on evolving domains. The physical domain is embedded into a fixed computational mesh such that arbitrary intersections of the moving domain's boundaries with the background mesh occur. The key ingredients of the approach are the weak formulation of Dirichlet boundary conditions by Nitsche's method, the flexible and efficient integration over all types of intersections of cells by moving boundaries and the spatial extension of the discrete physical quantities to the entire computational background mesh including fictitious (ghost) subdomains of fluid flow. To prevent spurious oscillations caused by irregular intersections of mesh cells, a penalization ensuring the stability of the approach and defining implicitly the extension to host domains is added. These techniques are embedded in an arbitrary order, discontinuous Galerkin discretization of the time variable and an inf-sup stable discretization of the spatial variables. The convergence and stability properties of the approach are studied, firstly, for a benchmark problem of flow around a stationary obstacle and, secondly, for flow around moving obstacles with arising cut cells and fictitious domains. The parallel implementation is also addressed.

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“…Here, γ 1 > 0 and γ 2 > 0 are numerical (tuning) parameters for the penalization. The linear form S F t h is introduced and analyzed for the Stokes problem in [3]. Its impact is twofold.…”

mentioning

confidence: 99%

“…Here, E denotes the canonical patchwise extension of the discrete functions. We refere to [1,3] for its definition. Putting A s h := A h + S F t h and applying a discontinuous Galerkin time discretization with piecewise polynomials of order k (cf.…”

mentioning

confidence: 99%

Numerical convergence of discrete extensions in a space‐time finite element, fictitious domain method for the Navier–Stokes equations

Anselmann

Bause

2021

Proc Appl Math & Mech

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A key ingredient of our fictitious domain, higher order space-time cut finite element (CutFEM) approach for solving the incompressible Navier-Stokes equations on evolving domains (cf. [1]) is the extension of the physical solution from the timedependent flow domain Ω t f to the entire, time-independent computational domain Ω. The extension is defined implicitly and, simultaneously, aims at stabilizing the discrete solution in the case of unavoidable irregular small cuts. Here, the convergence properties of the scheme are studied numerically for variations of the combined extension and stabilization.

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An unfitted Eulerian finite element method for the time-dependent Stokes problem on moving domains

Cited by 19 publications

References 43 publications

An evolving space framework for Oseen equations on a moving domain

An evolving space framework for Oseen equations on a moving domain

Cut finite element methods and ghost stabilization techniques for space‐time discretizations of the Navier–Stokes equations

Numerical convergence of discrete extensions in a space‐time finite element, fictitious domain method for the Navier–Stokes equations

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