A geometric multigrid method for space-time finite element discretizations of the Navier-Stokes equations and its application to 3d flow simulation

Markus Bause; Mathias Anselmann

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A geometric multigrid method for space-time finite element discretizations of the Navier-Stokes equations and its application to 3d flow simulation

Mathias Anselmann, Markus Bause

Numerical Analysis · Submitted 22 Jul 2021

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Its performance properties are demonstrated for 2d and 3d benchmarks of flow around a cylinder.

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arXiv ID2107.10561 (version 2)

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CitationMarkus Bause, Mathias Anselmann. "A geometric multigrid method for space-time finite element discretizations of the Navier-Stokes equations and its application to 3d flow simulation." arXiv:2107.10561 (2021). https://pith.science/paper/HT4QGK5L

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.14 · human participants 0.02 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.09 · mentions code or data availability 0.05.

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We present a parallelized geometric multigrid (GMG) method, based on the cell-based Vanka smoother, for higher order space-time finite element methods (STFEM) to the incompressible Navier--Stokes equations. The STFEM is implemented as a time marching scheme. The GMG solver is applied as a preconditioner for GMRES iterations. Its performance properties are demonstrated for 2d and 3d benchmarks of flow around a cylinder. The key ingredients of the GMG approach are the construction of the local Vanka smoother over all degrees of freedom in time of the respective subinterval and its efficient application. For this, data structures that store pre-computed cell inverses of the Jacobian for all hierarchical levels and require only a reasonable amount of memory overhead are generated. The GMG method is built for the \emph{deal.II} finite element library. The concepts are flexible and can be transferred to similar software platforms.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present a parallelized geometric multigrid (GMG) method, based on the cell-based Vanka smoother, for higher order space-time finite element methods (STFEM) to the incompressible Navier--Stokes equations.	contribution 0.98	0.94	0.21	asserted 1.00
s1	The STFEM is implemented as a time marching scheme.	method 0.98	0.80	0.03	asserted 1.00
s2	The GMG solver is applied as a preconditioner for GMRES iterations.	method 0.98	0.81	0.07	asserted 1.00
s3	Its performance properties are demonstrated for 2d and 3d benchmarks of flow around a cylinder.	result 0.77	0.93	0.19	asserted 1.00
s4	The key ingredients of the GMG approach are the construction of the local Vanka smoother over all degrees of freedom in time of the respective subinterval and its efficient application.	method 0.86	0.84	0.13	asserted 1.00
s5	For this, data structures that store pre-computed cell inverses of the Jacobian for all hierarchical levels and require only a reasonable amount of memory overhead are generated.	method 0.98	0.89	0.04	asserted 1.00
s6	The GMG method is built for the \emph{deal.II} finite element library.	method 0.80	0.80	0.03	asserted 1.00
s7	The concepts are flexible and can be transferred to similar software platforms.	interpretation 0.49	0.37	0.06	asserted 0.59

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 1.00.

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Original abstract

We present a parallelized geometric multigrid (GMG) method, based on the cell-based Vanka smoother, for higher order space-time finite element methods (STFEM) to the incompressible Navier--Stokes equations. The STFEM is implemented as a time marching scheme. The GMG solver is applied as a preconditioner for GMRES iterations. Its performance properties are demonstrated for 2d and 3d benchmarks of flow around a cylinder. The key ingredients of the GMG approach are the construction of the local Vanka smoother over all degrees of freedom in time of the respective subinterval and its efficient application. For this, data structures that store pre-computed cell inverses of the Jacobian for all hierarchical levels and require only a reasonable amount of memory overhead are generated. The GMG method is built for the \emph{deal.II} finite element library. The concepts are flexible and can be transferred to similar software platforms.

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3 arXiv papers cite this in 21 sentences

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Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

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.

Efficiency of local Vanka smoother geometric multigrid preconditioning for space-time finite element methods to the Navier-Stokes equations · 0.83

Papers citing it · 16 sentences without a confident label

Uses its method or tool 2 papers

For details of the parallel implementation of the geometric multigrid solver we refer to [ref] .

An energy-efficient GMRES-Multigrid solver for space-time finite element computation of dynamic poro- and thermoelasticity · Numerical studies · 0.80
The implementation of the numerical scheme was done in an in-house high-performance frontend solver for the library [ref] .

Convergence of a continuous Galerkin method for hyperbolic-parabolic systems · Numerical convergence test · 0.99

Compares with it 1 paper

This is in contrast to the Navier–Stokes equations for which elementwise Vanka-type smoother are known to perform excellently if discontinuous pressure approximations are applied; cf. [ref] .

An energy-efficient GMRES-Multigrid solver for space-time finite element computation of dynamic poro- and thermoelasticity · Space-time finite element and multigrid techniques · 0.80

Cites it as support 1 paper

Numerical evidence of these performance properties is presented in, e.g., [ref] .

Efficiency of local Vanka smoother geometric multigrid preconditioning for space-time finite element methods to the Navier-Stokes equations · Introduction · 0.99

Cites it as background 1 paper

For this we refer to, e.g., [ref] and the references therein.

Efficiency of local Vanka smoother geometric multigrid preconditioning for space-time finite element methods to the Navier-Stokes equations · Introduction · 0.94

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