Global and local pointwise error estimates for finite element approximations to the Stokes problem on convex polyhedra

Boris Vexler; Dmitriy Leykekhman; Niklas Behringer

arxiv: 1907.06871 · v1 · pith:EX6ENUHNnew · submitted 2019-07-16 · 🧮 math.NA · cs.NA

Global and local pointwise error estimates for finite element approximations to the Stokes problem on convex polyhedra

Niklas Behringer , Dmitriy Leykekhman , Boris Vexler This is my paper

Pith reviewed 2026-05-24 21:04 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords finite element methodStokes problempointwise error estimatesstability estimatesGreen's functionsconvex polyhedraW1 infinity normL infinity norm

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The pith

Finite element solutions of the Stokes problem achieve new pointwise stability and localization in W^{1,∞} and L^∞ norms on convex polyhedra.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes stability and localization results for finite element solutions of the Stokes equations in maximum norms. It extends stability estimates from regularized Green's functions to unstructured quasi-uniform meshes in two and three dimensions. These results provide interior error estimates that were previously known for elliptic problems but new for the Stokes system. A sympathetic reader would care because such estimates allow for better understanding of local behavior and pointwise accuracy without relying on structured meshes.

Core claim

The paper shows new stability and localization results for the finite element solution of the Stokes system in W^{1,∞} and L^∞ norms under standard assumptions on the finite element spaces on quasi-uniform meshes in two and three dimensions by extending previously known stability estimates using regularized Green's functions.

What carries the argument

Regularized Green's functions that extend stability estimates for the Stokes system to unstructured quasi-uniform meshes.

Load-bearing premise

The extension of previously known stability estimates for the Stokes system using regularized Green's functions is valid for the unstructured quasi-uniform meshes on convex polyhedra considered.

What would settle it

A specific finite element space and quasi-uniform mesh on a convex polyhedron where the W^{1,∞} or L^∞ stability estimate fails for the Stokes solution.

read the original abstract

The main goal of the paper is to show new stability and localization results for the finite element solution of the Stokes system in $W^{1,\infty}$ and $L^{\infty}$ norms under standard assumptions on the finite element spaces on quasi-uniform meshes in two and three dimensions. Although interior error estimates are well-developed for the elliptic problem, they appear to be new for the Stokes system on unstructured meshes. To obtain these results we extend previously known stability estimates for the Stokes system using regularized Green's functions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims new interior W^{1,∞} and L^∞ error estimates for Stokes FEM on unstructured quasi-uniform meshes by extending regularized Green's functions, but the abstract supplies no details on whether that extension actually works without extra mesh assumptions.

read the letter

The main thing to know is that this paper targets new pointwise stability and localization results for the finite element solution of the Stokes system in W^{1,∞} and L^∞ norms. It does so on convex polyhedra with standard finite element spaces and quasi-uniform meshes in two and three dimensions, and it flags the interior estimates on unstructured meshes as new compared with the elliptic case. The approach is to extend earlier stability estimates via regularized Green's functions. That is the core claim extracted from the abstract. The work is aimed squarely at people who need a priori pointwise control for incompressible flow discretizations, and the abstract is straightforward about separating the new Stokes content from prior elliptic results. It earns credit for stating the setting cleanly and for identifying a concrete gap rather than re-deriving known elliptic bounds. The soft spot is exactly the one the stress-test note flags: the abstract gives no technical steps on how the Green's function construction or the ensuing bounds adapt when the mesh is unstructured. Without those steps it is impossible to tell whether the extension really holds under only the stated quasi-uniformity assumption or whether it quietly needs more regularity that earlier papers supplied. If the proofs turn out to lean on hidden structure, the central claim would need revision. This is a narrow but well-defined contribution in numerical analysis of Stokes problems. A reader who works on pointwise estimates or who maintains fluid codes that rely on such bounds could extract value once the derivations are checked. It shows honest engagement with the literature by building explicitly on known elliptic techniques. I would bring the full paper to a reading group to walk through the Green's function construction. I would not cite it in my own work until the extension is verified. It deserves peer review because the claim is specific, the setting is standard, and the only way to settle the extension question is to see the proofs.

Referee Report

0 major / 3 minor

Summary. The paper claims to establish new global and local pointwise error estimates in the W^{1,∞} and L^∞ norms for finite element approximations to the Stokes problem on convex polyhedra. These results are obtained under standard assumptions on the finite element spaces and for quasi-uniform meshes in two and three dimensions by extending previously known stability estimates that employ regularized Green's functions. The estimates are presented as novel for the Stokes system on unstructured meshes, in contrast to existing interior estimates for elliptic problems.

Significance. If the extension of the regularized Green's function stability estimates is carried through rigorously, the results would provide useful new tools for pointwise and localized error analysis of Stokes finite element methods on general convex domains. This fills a gap relative to the elliptic case and could support the development of maximum-norm a posteriori estimators for incompressible flow problems. The manuscript explicitly builds on prior stability results rather than deriving everything from scratch, which is a strength.

minor comments (3)

[Abstract] The abstract states the results are obtained 'under standard assumptions on the finite element spaces on quasi-uniform meshes' but does not name the precise inf-sup stable pairs considered; adding this in §1 or §2 would improve clarity.
[Section 3] Notation for the regularized Green's function (likely introduced in §3) should be cross-referenced explicitly when the extension argument begins, to help readers track the adaptation from prior literature.
[Section 4] A brief remark on how the convex-polyhedron assumption enters the Green's function estimates (e.g., via boundary regularity) would strengthen the exposition without altering the main argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The summary correctly identifies the extension of regularized Green's function techniques to obtain new global and local pointwise estimates in W^{1,∞} and L^∞ norms for the Stokes system on convex polyhedra under standard assumptions on quasi-uniform meshes.

Circularity Check

0 steps flagged

No significant circularity; results obtained by extending prior stability estimates

full rationale

The paper derives its W^{1,∞} and L^∞ stability and localization results for the Stokes finite-element problem by extending previously known stability estimates that rely on regularized Green's functions. This extension is performed under the stated assumptions on the finite-element spaces and quasi-uniform meshes, with the abstract and description explicitly framing the contribution as this extension rather than any self-referential definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. No equations or steps reduce the claimed results to the inputs by construction, and the derivation remains self-contained against external benchmarks from prior literature.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only abstract available, so ledger is limited to explicitly mentioned assumptions; no free parameters, invented entities, or detailed axioms visible.

axioms (2)

domain assumption Standard assumptions on the finite element spaces
Invoked as basis for the stability and localization results.
domain assumption Quasi-uniform meshes in two and three dimensions on convex polyhedra
Required for the new estimates to hold.

pith-pipeline@v0.9.0 · 5616 in / 1144 out tokens · 20350 ms · 2026-05-24T21:04:57.877248+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

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J. C. de los Reyes, C. Meyer, and B. Vexler , Finite element error analysis for state- constrained optimal control of the Stokes equations , Control Cybernet., 37 (2008), pp. 251– 284

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V. Girault, R. H. Nochetto, and L. R. Scott , Max-norm estimates for Stokes and Navier- Stokes approximations in convex polyhedra , Numer. Math., 131 (2015), pp. 771–822

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Guzm ´an and D

J. Guzm ´an and D. Leykekhman , Pointwise error estimates of ﬁnite element approximations to the Stokes problem on convex polyhedra , Math. Comp., 81 (2012), pp. 1879–1902, https:// doi.org/10.1090/S0025-5718-2012-02603-2

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Leykekhman and M

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work page doi:10.1090/mcom/3117 2017
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Leykekhman and B

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Leykekhman and B

D. Leykekhman and B. Vexler , Pointwise best approximation results for Galerkin ﬁnite element solutions of parabolic problems , SIAM J. Numer. Anal., 54 (2016), pp. 1365–1384, https://doi.org/10.1137/15M103412X

work page doi:10.1137/15m103412x 2016
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Leykekhman and B

D. Leykekhman and B. Vexler , Discrete maximal parabolic regularity for Galerkin ﬁ- nite element methods , Numer. Math., 135 (2017), pp. 923–952, https://doi.org/1 0.1007/ s00211-016-0821-2

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Leykekhman and B

D. Leykekhman and B. Vexler , Global and interior pointwise best approximation results for the gradient of Galerkin solutions for parabolic proble ms, SIAM J. Numer. Anal., 55 (2017), pp. 2025–2049, https://doi.org/10.1137/16M1080 252

work page doi:10.1137/16m1080 2017
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Maz’ya and B

V. Maz’ya and B. Plamenevskii , The ﬁrst boundary value problem for classical equations of mathematical physics in domains with piecewise-smooth bou ndaries. I , Z. Anal. Anwen- dungen, 2 (1983), pp. 335–359, https://doi.org/10.4171/Z AA/71

work page doi:10.4171/z 1983
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Maz’ya and J

V. Maz’ya and J. Rossmann , Lp estimates of solutions to mixed boundary value problems for the Stokes system in polyhedral domains , Math. Nachr., 280 (2007), pp. 751–793, https:// doi.org/10.1002/mana.200610513

work page doi:10.1002/mana.200610513 2007
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Maz’ya and J

V. Maz’ya and J. Rossmann , Elliptic equations in polyhedral domains , vol. 162 of Mathe- matical Surveys and Monographs, American Mathematical Soc iety, Providence, RI, 2010, https://doi.org/10.1090/surv/162

work page doi:10.1090/surv/162 2010
[22]

Narasimhan and I

R. Narasimhan and I. Babuska , Interior maximum norm estimates for ﬁnite element dis- cretizations of the Stokes equations , Appl. Anal., 86 (2007), pp. 251–260, https://doi.org/ 10.1080/00036810601148240

work page doi:10.1080/00036810601148240 2007
[23]

Natterer , ¨Uber die punktweise Konvergenz ﬁniter Elemente , Numer

F. Natterer , ¨Uber die punktweise Konvergenz ﬁniter Elemente , Numer. Math., 25 (1974), pp. 67–77, https://doi.org/10.1007/BF01419529

work page doi:10.1007/bf01419529 1974
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Pieper and B

K. Pieper and B. Vexler , A priori error analysis for discretization of sparse ellipt ic optimal control problems in measure space , SIAM J. Control Optim., 51 (2013), pp. 2788–2808, https://doi.org/10.1137/120889137

work page doi:10.1137/120889137 2013
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Rannacher , Zur L∞-Konvergenz linearer ﬁniter Elemente beim Dirichlet-Prob lem, Math

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A. H. Schatz , A weak discrete maximum principle and stability of the ﬁnite element method in L∞ on plane polygonal domains. I , Math. Comp., 34 (1980), pp. 77–91, https://doi. org/10.2307/2006221

work page doi:10.2307/2006221 1980
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A. H. Schatz and L. B. W ahlbin , Interior maximum norm estimates for ﬁnite element methods, Math. Comp., 31 (1977), pp. 414–442, https://doi.org/10. 2307/2006424

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[29]

A. H. Schatz and L. B. W ahlbin , Interior maximum-norm estimates for ﬁnite element meth- ods. II , Math. Comp., 64 (1995), pp. 907–928, https://doi.org/10. 2307/2153476

work page 1995

[1] [1]

D. N. Arnold and X. Liu , Local error estimates for ﬁnite element discretizations of the Stokes equations, RAIRO Mod´ el. Math. Anal. Num´ er., 29 (1995), pp. 367–389, https://doi.org/ 10.1051/m2an/1995290303671

work page doi:10.1051/m2an/1995290303671 1995

[2] [2]

Chen , Pointwise error estimates for ﬁnite element solutions of th e Stokes problem , SIAM J

H. Chen , Pointwise error estimates for ﬁnite element solutions of th e Stokes problem , SIAM J. Numer. Anal., 44 (2006), pp. 1–28, https://doi.org/10.1 137/S0036142903438100

work page 2006

[3] [3]

Dauge, Stationary Stokes and Navier-Stokes systems on two- or thre e-dimensional domains with corners

M. Dauge, Stationary Stokes and Navier-Stokes systems on two- or thre e-dimensional domains with corners. I. Linearized equations , SIAM J. Math. Anal., 20 (1989), pp. 74–97

work page 1989

[4] [4]

J. C. de los Reyes and K. Kunisch , A semi-smooth Newton method for control constrained boundary optimal control of the Navier-Stokes equations , Nonlinear Anal., 62 (2005), pp. 1289–1316, https://doi.org/10.1016/j.na.2005.04.0 35

work page doi:10.1016/j.na.2005.04.0 2005

[5] [6]

J. C. de los Reyes, C. Meyer, and B. Vexler , Finite element error analysis for state- constrained optimal control of the Stokes equations , Control Cybernet., 37 (2008), pp. 251– 284

work page 2008

[6] [7]

R. G. Dur ´an and M. A. Muschietti , An explicit right inverse of the divergence operator which is continuous in weighted norms , Studia Math., 148 (2001), pp. 207–219. 24 N. BEHRINGER, D. LEYKEKHMAN AND B. VEXLER

work page 2001

[7] [8]

R. G. Dur ´an, R. H. Nochetto, and J. P. W ang , Sharp maximum norm error estimates for ﬁnite element approximations of the Stokes problem in 2-D, Math. Comp., 51 (1988), pp. 491–506, https://doi.org/10.2307/2008760

work page doi:10.2307/2008760 1988

[8] [9]

Ern and J

A. Ern and J. Guermond , Theory and practice of ﬁnite elements , vol. 159 of Ap- plied Mathematical Sciences, Springer-Verlag, New York, 2 004, https://doi.org/10.1007/ 978-1-4757-4355-5

work page

[9] [10]

G. P. Galdi , An introduction to the mathematical theory of the Navier-St okes equations , Springer Monographs in Mathematics, Springer, New York, se cond ed., 2011, https://doi. org/10.1007/978-0-387-09620-9. Steady-state problems

work page doi:10.1007/978-0-387-09620-9 2011

[10] [11]

Girault, R

V. Girault, R. H. Nochetto, and L. R. Scott , Maximum-norm stability of the ﬁnite element Stokes projection , J. Math. Pures Appl. (9), 84 (2005), pp. 279–330, https://d oi.org/10. 1016/j.matpur.2004.09.017

work page 2005

[11] [12]

Girault, R

V. Girault, R. H. Nochetto, and L. R. Scott , Max-norm estimates for Stokes and Navier- Stokes approximations in convex polyhedra , Numer. Math., 131 (2015), pp. 771–822

work page 2015

[12] [13]

Guzm ´an and D

J. Guzm ´an and D. Leykekhman , Pointwise error estimates of ﬁnite element approximations to the Stokes problem on convex polyhedra , Math. Comp., 81 (2012), pp. 1879–1902, https:// doi.org/10.1090/S0025-5718-2012-02603-2

work page doi:10.1090/s0025-5718-2012-02603-2 2012

[13] [14]

Leykekhman and M

D. Leykekhman and M. Pruitt , On the positivity of discrete harmonic functions and the discrete Harnack inequality for piecewise linear ﬁnite ele ments, Math. Comp., 86 (2017), pp. 1127–1145, https://doi.org/10.1090/mcom/3117

work page doi:10.1090/mcom/3117 2017

[14] [15]

Leykekhman and B

D. Leykekhman and B. Vexler , Finite element pointwise results on convex polyhedral domains, SIAM J. Numer. Anal., 54 (2016), pp. 561–587, https://doi. org/10.1137/ 15M1013912

work page 2016

[15] [16]

Leykekhman and B

D. Leykekhman and B. Vexler , Pointwise best approximation results for Galerkin ﬁnite element solutions of parabolic problems , SIAM J. Numer. Anal., 54 (2016), pp. 1365–1384, https://doi.org/10.1137/15M103412X

work page doi:10.1137/15m103412x 2016

[16] [17]

Leykekhman and B

D. Leykekhman and B. Vexler , Discrete maximal parabolic regularity for Galerkin ﬁ- nite element methods , Numer. Math., 135 (2017), pp. 923–952, https://doi.org/1 0.1007/ s00211-016-0821-2

work page 2017

[17] [18]

Leykekhman and B

D. Leykekhman and B. Vexler , Global and interior pointwise best approximation results for the gradient of Galerkin solutions for parabolic proble ms, SIAM J. Numer. Anal., 55 (2017), pp. 2025–2049, https://doi.org/10.1137/16M1080 252

work page doi:10.1137/16m1080 2017

[18] [19]

Maz’ya and B

V. Maz’ya and B. Plamenevskii , The ﬁrst boundary value problem for classical equations of mathematical physics in domains with piecewise-smooth bou ndaries. I , Z. Anal. Anwen- dungen, 2 (1983), pp. 335–359, https://doi.org/10.4171/Z AA/71

work page doi:10.4171/z 1983

[19] [20]

Maz’ya and J

V. Maz’ya and J. Rossmann , Lp estimates of solutions to mixed boundary value problems for the Stokes system in polyhedral domains , Math. Nachr., 280 (2007), pp. 751–793, https:// doi.org/10.1002/mana.200610513

work page doi:10.1002/mana.200610513 2007

[20] [21]

Maz’ya and J

V. Maz’ya and J. Rossmann , Elliptic equations in polyhedral domains , vol. 162 of Mathe- matical Surveys and Monographs, American Mathematical Soc iety, Providence, RI, 2010, https://doi.org/10.1090/surv/162

work page doi:10.1090/surv/162 2010

[21] [22]

Narasimhan and I

R. Narasimhan and I. Babuska , Interior maximum norm estimates for ﬁnite element dis- cretizations of the Stokes equations , Appl. Anal., 86 (2007), pp. 251–260, https://doi.org/ 10.1080/00036810601148240

work page doi:10.1080/00036810601148240 2007

[22] [23]

Natterer , ¨Uber die punktweise Konvergenz ﬁniter Elemente , Numer

F. Natterer , ¨Uber die punktweise Konvergenz ﬁniter Elemente , Numer. Math., 25 (1974), pp. 67–77, https://doi.org/10.1007/BF01419529

work page doi:10.1007/bf01419529 1974

[23] [24]

Pieper and B

K. Pieper and B. Vexler , A priori error analysis for discretization of sparse ellipt ic optimal control problems in measure space , SIAM J. Control Optim., 51 (2013), pp. 2788–2808, https://doi.org/10.1137/120889137

work page doi:10.1137/120889137 2013

[24] [25]

Rannacher , Zur L∞-Konvergenz linearer ﬁniter Elemente beim Dirichlet-Prob lem, Math

R. Rannacher , Zur L∞-Konvergenz linearer ﬁniter Elemente beim Dirichlet-Prob lem, Math. Z., 149 (1976), pp. 69–77, https://doi.org/10.1007/BF013 01633

work page doi:10.1007/bf013 1976

[25] [26]

Rossmann , Green ’s matrix of the Stokes system in a convex polyhedron , Rostock

J. Rossmann , Green ’s matrix of the Stokes system in a convex polyhedron , Rostock. Math. Kolloq., (2010), pp. 15–28

work page 2010

[26] [27]

A. H. Schatz , A weak discrete maximum principle and stability of the ﬁnite element method in L∞ on plane polygonal domains. I , Math. Comp., 34 (1980), pp. 77–91, https://doi. org/10.2307/2006221

work page doi:10.2307/2006221 1980

[27] [28]

A. H. Schatz and L. B. W ahlbin , Interior maximum norm estimates for ﬁnite element methods, Math. Comp., 31 (1977), pp. 414–442, https://doi.org/10. 2307/2006424

work page 1977

[28] [29]

A. H. Schatz and L. B. W ahlbin , Interior maximum-norm estimates for ﬁnite element meth- ods. II , Math. Comp., 64 (1995), pp. 907–928, https://doi.org/10. 2307/2153476

work page 1995