Eciency of local Vanka smoother geometric multigrid preconditioning for space-time nite element methods to the NavierStokes

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Eﬃciency of local Vanka smoother geometric

multigrid preconditioning for space-time ﬁnite

element methods to the Navier–Stokes

equations

Mathias Anselmann†, Markus Bause‡∗

†,‡Helmut Schmidt University, Faculty of Mechanical and Civil Engineering, Holstenhofweg 85,

22043 Hamburg, Germany

Numerical simulation of incompressible viscous ﬂow, in particular in three space dimen-

sions, continues to remain a challenging task. 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. Linearizing the resulting al-

gebraic problems by Newton’s method yields linear systems with block matrices built of

(k+1)×(k+1)saddle point systems, where kdenotes 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 bench-

mark problem of ﬂow around a cylinder. Here, the robustness of the solver with respect to

the piecewise polynomial order kin time is analyzed and proved numerically.

1 Introduction

Let Ω ⊂Rdwith d=2,3 denote a bounded Lipschitz domain and I∶=(0, T ]for some ﬁnal time T>0.

We consider the numerical approximation by space-time ﬁnite element methods of solutions to the

nonstationary Navier–Stokes system

∂tv+(v⋅∇)v−ν∆v+∇p=f,∇⋅v=0 in Ω ×T , (1.1)

equipped with the initial condition v(0)=v0in Ω and appropriate boundary conditions on the diﬀerent

parts of ∂Ω, either the Dirichlet condition v=gor the do-nothing condition T(v, p)⋅n=0with the

asymmetric stress tensor T(v, p)=ν∇v−pIand the outer unit normal vector n. In Section 4, the

benchmark setting of ﬂow around a cylinder (cf. [18]) is chosen as Ω.

In the recent decade, the development and analysis of space-time ﬁnite element methods (STFEMs)

have strongly attracted researchers’ interest. For this we refer to, e.g., [1–4, 11] and the references

therein. STFEMs oﬀer appreciable advantages like the natural construction of higher order schemes,

the application of duality based concepts of a-posteriori error control and space-time mesh adaptivity [5]

and the natural discretization of coupled problems of multi-physics. Here, discontinuous Galerkin

time discretizations and inf-sup stable ﬁnite element methods in space with discontinuous pressure

∗bause@hsu-hh.de (corresponding author)

arXiv:2210.02690v1 [math.NA] 6 Oct 2022

approximation are employed. For the time variable, a Lagrangian basis with respect to the (k+1)

Gauss–Radau quadrature points of each subinterval is used. By the choice of a discontinuous temporal

test basis, a time-marching scheme is obtained, with temporal degrees of freedom associated with the

Gauss–Radau quadrature nodes of the subintervals. For details of the construction of the schemes we

refer to [2, 3]. We linearize the resulting nonlinear algebraic problem by a damped version of Newton’s

method. This yields linear systems of equations with block matrices of (k+1)×(k+1)subsystems, with

each of them having saddle point structure. Here, kdenotes the piecewise polynomial order of the time

discretization; cf. [2,4,11]. The structure of each of the subsystems resembles the discretization of (1.1)

by the implicit Euler method in time and inf-sup stable pairs of ﬁnite elements in space. We note that

there exits a strong link between the implicit Euler and the lowest order discontinuous Gakerkin time

discretization; cf. [20]. The solution of the problems demands for eﬃcient and robust iteration schemes.

Geometric multigrid (GMG) methods are known as the most eﬃcient iterative techniques for the

solution of large linear systems arising from the discretization of partial diﬀerential equations, including

incompressible viscous ﬂow. GMG methods are applied in many variants [6, 10, 15, 19]. They exploit

diﬀerent mesh levels of the underlying problem in order to reduce diﬀerent frequencies of the error

by employing a relatively cheap smoother on each grid level. Diﬀerent iterative methods have been

proposed in the literature as smoothing procedure. They range from low-cost methods like Richardson,

Jacobi, and SOR applied to the normal equation of the linear system to collective smoothers, that

are based on the solution of small local problems. Here, we use a Vanka-type smoother [16, 21] of

the family of collective methods. Nowadays, GMG methods are employed as preconditioner in Krylov

subspace iterations, for instance GMRES iterations, to enhance the linear solver’s robustness, which

is also done here. Parallel implementations of GMG techniques on modern computer architectures

show excellent scalability properties and their high eﬃciency has been recognized. Numerical evidence

of these performance properties is presented in, e.g., [2, 7, 8, 14, 15]. Analyses of GMG methods (cf.,

e.g., [6,10,16]) have been done for linear systems in saddle point form, with matrix A=A B⊺

B−Cand

symmetric and positive deﬁnite submatrices Aand C, arising for instance from mixed discretizations

of the Stokes problem. If higher order discontinuous Galerkin time stepping is used, the resulting linear

system matrix is a (k+1)×(k+1)block matrix with each block being of the form A. This imposes

an additional facet of complexity on the geometric multigrid preconditioner. In this work, we study

numerically the performance properties of GMRES iterations with GMG preconditioning for space-

time ﬁnite element discretizations of (1.1) . In particular, higher order polynomial approximation of

the temporal variable is investigated. The benchmark of ﬂow around a cylinder [18] in two- and three

space dimensions is chosen as a test problem. Beyond the expectable spatial grid independency of the

number of iterations of the linear solver, that is already shown numerically in [3,11], the robustness of

the solver with respect to the (piecewise) polynomial degree of the time discretization is analyzed and

shown here.

2 Numerical scheme

For the time discretization, we decompose I=(0, T ]into Nsubintervals In=(tn−1, tn],n=1,...,N,

where 0 =t0<t1<⋯<tN−1<tN=T. For the space discretization, let {Tl}L

l=0be the decomposition on

every multigrid level of Ω into (open) quadrilaterals or hexahedrals, with Tl={Kii=1,...,Nel

l}, for

l=0,...,L. The ﬁnest partition is Th=TL. We assume that all the partitions {Tl}L

l=0are quasi-uniform

with characteristic mesh size hland hl=γhl−1,γ∈(0,1)and h0=O(1). On the actual mesh level,

we denote by Vh×Qh⊂H1(Ω)×L2(Ω)the ﬁnite element spaces that are built on the inf-sup stable

pair of spaces (Qr)d×Pdisc

r−1for some r≥2; cf. [13, p. 115]. By an abuse of notation, we skip the index

lof the mesh level when it is clear from the context. We let R∶=dim Vhand S∶=dim Qhas well as

R=dim((Qr)d)and ˆ

S=dim(Pdisc

r−1). For the application of Dirichlet boundary conditions we employ

Nitsche’s method. This is only done since our approach and its implementation are embedded in a more

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EciencyoflocalVankasmoothergeometricmultigridpreconditioningforspace-timeniteelementmethodstotheNavier{StokesequationsMathiasAnselmann,MarkusBause*;HelmutSchmidtUniversity,FacultyofMechanicalandCivilEngineering,Holstenhofweg85,22043Hamburg,GermanyNumericalsimulationofincompressibleviscousow,in...

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Eciency of local Vanka smoother geometric multigrid preconditioning for space-time nite element methods to the NavierStokes

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