A Note on Accurate and Efficient Higher Order Galerkin Time Stepping Schemes for the Nonstationary Stokes Equations

Hussain, Shafqat

doi:10.2174/1876389801204010035

Cited by 26 publications

(36 citation statements)

References 8 publications

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“…Then, the initial condition is equivalent to the condition

{bold-italicU}_{n}^{0} MathClass-rel= {bold-italicu}_{τ} {MathClass-rel|}_{I_{n MathClass-bin- 1}} (t_{n MathClass-bin- 1}) if n ⩾ 2 or {bold-italicU}_{n}^{0} MathClass-rel= {bold-italicu}_{0} if n MathClass-rel= 1 MathClass-punc.

The other points

t_{n MathClass-punc, 1} MathClass-punc, MathClass-op\dots MathClass-punc, t_{n MathClass-punc, k}

are chosen as the quadrature points of the k ‐point Gaussian formula on I n which is exact if the function to be integrated is a polynomial of degree less or equal to 2 k − 1. We define the basis functions

φ_{n MathClass-punc, j} MathClass-rel\in {double-struckP}_{k} (I_{n})

of via affine reference transformations (see for more details). Now, we can describe the time discrete I n ‐problem of the cGP( k ) method :…”

Section: Galerkin Time‐stepping For the Navier–stokes Equationsmentioning

confidence: 99%

“…We use the one‐point Gaussian quadrature formula with

{\overset{t}{true}}_{1} MathClass-rel= 0

and

t_{n MathClass-punc, 1} MathClass-rel= t_{n MathClass-bin- 1} MathClass-bin+ \frac{τ_{n}}{2}

. Then, we get α 1,0 = − 1 and α 1,1 = 1 . Thus, problem leads to the following problem for the ‘one’ pair of unknowns

{bold-italicU}_{n}^{1} MathClass-rel= {bold-italicu}_{τ} (t_{n MathClass-bin- 1} MathClass-bin+ \frac{τ_{n}}{2})

and

P_{n}^{1} MathClass-rel= p_{τ} (t_{n MathClass-bin- 1} MathClass-bin+ \frac{τ_{n}}{2})

: Find

({bold-italicU}_{n}^{1} MathClass-punc, P_{n}^{1}) MathClass-rel\in bold-italicV MathClass-bin\times Q

such that for all ( v , q ) ∈ V × Q , it holds

\begin{matrix} falsenone none nonefalsearray \\ arraybaseline {(U_{n}^{1}, v)}_{Ω} + \frac{τ_{n}}{2} a (U_{n}^{1}, v) + \frac{τ_{n}}{2} n (U_{n}^{1}, U_{n}^{1}, v) +... \end{matrix}

…”

Section: Galerkin Time‐stepping For the Navier–stokes Equationsmentioning

confidence: 99%

“…In this paper, we will concentrate only on the case k = 2, that is, on the well‐known dG(1) method. We can derive the following constants for i , j ∈ {1,2} and

(α_{i MathClass-punc, j}) MathClass-rel= ((), \begin{matrix} falsenonefalsearray \\ arraycenter 1 & arraycenter \frac{\sqrt{3} - 1}{2} \\ arraycenter \frac{- \sqrt{3} - 1}{2} & arraycenter 1 \end{matrix}) MathClass-punc, (d_{i}) MathClass-rel= ((), \begin{matrix} falsenonefalsearray \\ arraycenter \frac{\sqrt{3} + 1}{2} \\ arraycenter \frac{- \sqrt{3} + 1}{2} \end{matrix}) MathClass-punc.

Then, on the time interval I n , one has to determine the two ‘unknowns’

({bold-italicU}_{n}^{j} MathClass-punc, P_{n}^{j}) MathClass-rel\in bold-italicV MathClass-bin\times Q

as the solution of the following coupled system

\begin{matrix} falsenone nonefalsearray \\ arraybaseline α_{1, 1} {(U_{n}^{1}, v)}_{Ω} + \frac{τ_{n}}{2} a (U_{n}^{1}, v) + \frac{τ_{n}}{2} n (U_{n}^{1}, U_{n}^{1}, v) + \frac{τ_{n}}{2 ...} \end{matrix}

…”

Section: Galerkin Time‐stepping For the Navier–stokes Equationsmentioning

confidence: 99%

“…In this paper, we extend our work in on the Galerkin‐type space–time discretizations dG( k ) and cGP( k ) to the case of the nonstationary Navier–Stokes equations where k denotes the degree of the time polynomial ansatz. We do not address the topic of adaptivity and assume that the spatial mesh is fixed on all time slabs.…”

Section: Introductionmentioning

confidence: 99%

“…It is also desirable to get a superconvergent pressure at the same time points, for instance, for the computation of the hydrodynamic forces in CFD problems such as drag, lift etc. Therefore, we perform additionally a special post‐processing based on a simple local interpolation procedure as described in . Our corresponding spatial discretization is based on using (bi/tri)quadratic finite elements ( Q 2 ) for the velocity and discontinuous linear elements (

P_{1}^{disc}

) for the pressure which are of similar higher order accuracy in space, namely of third, resp., second order measured in the L 2 ‐norm.…”

Section: Introductionmentioning

confidence: 99%

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An efficient and stable finite element solver of higher order in space and time for nonstationary incompressible flow

Hussain

Schieweck

Turek

2013

Numerical Methods in Fluids

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SUMMARYIn this paper, we present fully implicit continuous Galerkin–Petrov (cGP) and discontinuous Galerkin (dG) time‐stepping schemes for incompressible flow problems which are, in contrast to standard approaches like for instance the Crank–Nicolson scheme, of higher order in time. In particular, we analyze numerically the higher order dG(1) and cGP(2) methods, which are super convergent of third, resp., fourth order in time, whereas for the space discretization, the well‐known LBB‐stable finite element pair Q2MathClass-bin∕P1disc of third‐order accuracy is used. The discretized systems of nonlinear equations are treated by using the Newton method, and the associated linear subproblems are solved by means of a monolithic (geometrical) multigrid method with a blockwise Vanka‐like smoother treating all components simultaneously. We perform nonstationary simulations (in 2D) for two benchmarking configurations to analyze the temporal accuracy and efficiency of the presented time discretization schemes w.r.t. CPU and numerical costs. As a first test problem, we consider a classical ‘flow around cylinder’ benchmark. Here, we concentrate on the nonstationary behavior of the flow patterns with periodic oscillations and examine the ability of the different time discretization schemes to capture the dynamics of the flow. As a second test case, we consider the nonstationary ‘flow through a Venturi pipe’. The objective of this simulation is to control the instantaneous and mean flux through this device. Copyright © 2013 John Wiley & Sons, Ltd.

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“…Then, the initial condition is equivalent to the condition

{bold-italicU}_{n}^{0} MathClass-rel= {bold-italicu}_{τ} {MathClass-rel|}_{I_{n MathClass-bin- 1}} (t_{n MathClass-bin- 1}) if n ⩾ 2 or {bold-italicU}_{n}^{0} MathClass-rel= {bold-italicu}_{0} if n MathClass-rel= 1 MathClass-punc.

The other points

t_{n MathClass-punc, 1} MathClass-punc, MathClass-op\dots MathClass-punc, t_{n MathClass-punc, k}

φ_{n MathClass-punc, j} MathClass-rel\in {double-struckP}_{k} (I_{n})

of via affine reference transformations (see for more details). Now, we can describe the time discrete I n ‐problem of the cGP( k ) method :…”

Section: Galerkin Time‐stepping For the Navier–stokes Equationsmentioning

confidence: 99%

“…We use the one‐point Gaussian quadrature formula with

{\overset{t}{true}}_{1} MathClass-rel= 0

and

t_{n MathClass-punc, 1} MathClass-rel= t_{n MathClass-bin- 1} MathClass-bin+ \frac{τ_{n}}{2}

. Then, we get α 1,0 = − 1 and α 1,1 = 1 . Thus, problem leads to the following problem for the ‘one’ pair of unknowns

{bold-italicU}_{n}^{1} MathClass-rel= {bold-italicu}_{τ} (t_{n MathClass-bin- 1} MathClass-bin+ \frac{τ_{n}}{2})

and

P_{n}^{1} MathClass-rel= p_{τ} (t_{n MathClass-bin- 1} MathClass-bin+ \frac{τ_{n}}{2})

: Find

({bold-italicU}_{n}^{1} MathClass-punc, P_{n}^{1}) MathClass-rel\in bold-italicV MathClass-bin\times Q

such that for all ( v , q ) ∈ V × Q , it holds

\begin{matrix} falsenone none nonefalsearray \\ arraybaseline {(U_{n}^{1}, v)}_{Ω} + \frac{τ_{n}}{2} a (U_{n}^{1}, v) + \frac{τ_{n}}{2} n (U_{n}^{1}, U_{n}^{1}, v) +... \end{matrix}

…”

Section: Galerkin Time‐stepping For the Navier–stokes Equationsmentioning

confidence: 99%

“…In this paper, we will concentrate only on the case k = 2, that is, on the well‐known dG(1) method. We can derive the following constants for i , j ∈ {1,2} and

(α_{i MathClass-punc, j}) MathClass-rel= ((), \begin{matrix} falsenonefalsearray \\ arraycenter 1 & arraycenter \frac{\sqrt{3} - 1}{2} \\ arraycenter \frac{- \sqrt{3} - 1}{2} & arraycenter 1 \end{matrix}) MathClass-punc, (d_{i}) MathClass-rel= ((), \begin{matrix} falsenonefalsearray \\ arraycenter \frac{\sqrt{3} + 1}{2} \\ arraycenter \frac{- \sqrt{3} + 1}{2} \end{matrix}) MathClass-punc.

Then, on the time interval I n , one has to determine the two ‘unknowns’

({bold-italicU}_{n}^{j} MathClass-punc, P_{n}^{j}) MathClass-rel\in bold-italicV MathClass-bin\times Q

as the solution of the following coupled system

\begin{matrix} falsenone nonefalsearray \\ arraybaseline α_{1, 1} {(U_{n}^{1}, v)}_{Ω} + \frac{τ_{n}}{2} a (U_{n}^{1}, v) + \frac{τ_{n}}{2} n (U_{n}^{1}, U_{n}^{1}, v) + \frac{τ_{n}}{2 ...} \end{matrix}

…”

Section: Galerkin Time‐stepping For the Navier–stokes Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

P_{1}^{disc}

) for the pressure which are of similar higher order accuracy in space, namely of third, resp., second order measured in the L 2 ‐norm.…”

Section: Introductionmentioning

confidence: 99%

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An efficient and stable finite element solver of higher order in space and time for nonstationary incompressible flow

Hussain

Schieweck

Turek

2013

Numerical Methods in Fluids

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Efficiency of local Vanka smoother geometric multigrid preconditioning for space‐time finite element methods to the Navier–Stokes equations

Anselmann

Bause

2023

Proc Appl Math and Mech

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Numerical simulation of incompressible viscous flow continues to remain a challenging task, in particular if three space dimensions are involved. Space-time finite element methods feature the natural construction of higher order discretization schemes. They offer the potential to achieve accurate results on computationally feasible grids. Using a temporal test basis supported on the subintervals and linearizing the resulting algebraic problems by Newton's method yield linear systems of equations with block matrices built of (k + 1) × (k + 1) saddle point systems, where k denotes the polynomial order of the variational time discretization. We demonstrate numerically the efficiency of preconditioning GMRES iterations for solving these linear systems by a V -cycle geometric multigrid approach based on a local Vanka smoother. The studies are done for the two-and three-dimensional benchmark problem of flow around a cylinder. The robustness of the solver with respect to the piecewise polynomial order k in time is analyzed and confirmed numerically.

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Numerical Studies of Higher Order Variational Time Stepping Schemes for Evolutionary Navier-Stokes Equations

Ahmed¹,

Matthies²

2017

Lecture Notes in Computational Science and Engineering

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A Note on Accurate and Efficient Higher Order Galerkin Time Stepping Schemes for the Nonstationary Stokes Equations

Cited by 26 publications

References 8 publications

An efficient and stable finite element solver of higher order in space and time for nonstationary incompressible flow

An efficient and stable finite element solver of higher order in space and time for nonstationary incompressible flow

Efficiency of local Vanka smoother geometric multigrid preconditioning for space‐time finite element methods to the Navier–Stokes equations

Numerical Studies of Higher Order Variational Time Stepping Schemes for Evolutionary Navier-Stokes Equations

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