arxiv: 2605.04786 · v1 · submitted 2026-05-06 · 🧮 math.NA · cs.NA

Recognition: unknown

Superconvergence in finite element method by smoothing

Han Shui, Ludmil Zikatanov, Yuwen Li

Pith reviewed 2026-05-08 16:22 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords superconvergencefinite element methodsmoothingpostprocessingadditive smoothersmultiplicative smootherssymmetric positive definite problems

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The pith

Embedding the finite element solution into an enriched space and applying a few smoothing iterations produces superconvergence for symmetric positive definite problems.

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

This paper introduces a postprocessing technique that takes the standard finite element solution, embeds it into a richer finite element space, and applies a few iterations of common smoothers such as damped Jacobi or Gauss-Seidel. The result is an improved approximation that converges faster than the original finite element solution. The authors prove this superconvergence for symmetric positive definite problems using both additive and multiplicative smoothing strategies. The method is algebraic, requires no special operators, and extends to high-order and three-dimensional cases, as shown by experiments on Poisson, Maxwell, biharmonic, and Helmholtz equations.

Core claim

For symmetric and positive-definite problems, embedding the current finite element solution into an enriched finite element space and performing a few smoothing iterations with additive or multiplicative smoothers produces a superconvergent solution whose error is asymptotically smaller than that of the original finite element approximation.

What carries the argument

The smoothing postprocessing procedure consisting of embedding the finite element solution into an enriched finite element space followed by a small number of damped Jacobi, Gauss-Seidel, or conjugate gradient iterations.

If this is right

The procedure applies directly to high-order and three-dimensional discretizations without constructing auxiliary meshes or recovery operators.
Superconvergence is established for both additive and multiplicative smoothers on symmetric positive definite problems.
Numerical tests confirm the gains for Poisson, Maxwell, biharmonic, and Helmholtz equations.
The method remains algebraic and easy to implement in existing finite element codes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This smoothing step could serve as a cheap way to obtain better a posteriori error estimates for adaptive refinement.
Similar smoothing might improve accuracy in time-dependent problems if applied after each time step.
Extensions to non-symmetric or indefinite problems would require different analysis but could be tested numerically first.

Load-bearing premise

That the embedding of the finite element solution into the enriched space combined with the smoothing iterations produces the superconvergence property for symmetric positive definite problems.

What would settle it

Computing the error of the smoothed solution on a sequence of refined meshes for a symmetric positive definite problem and observing that the convergence rate does not exceed the standard finite element rate would falsify the superconvergence claim.

Figures

Figures reproduced from arXiv: 2605.04786 by Han Shui, Ludmil Zikatanov, Yuwen Li.

read the original abstract

This paper develops a smoothing-based postprocessing method for superconvergence in finite element methods. The method applies a few smoothing iterations, such as damped Jacobi, Gauss-Seidel, or conjugate gradient, with initial guess being the current finite element solution embedded in an enriched finite element space. The resulting procedure is algebraic, easy to implement, and applicable to high-order and three-dimensional discretizations. For symmetric and positive-definite problems, we prove superconvergence of the smoothed solutions under additive and multiplicative smoothers. Effectiveness of the proposed method is demonstrated by numerical experiments for the Poisson, Maxwell, biharmonic and Helmholtz equations.

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

Embedding the finite element solution in an enriched space and running a few smoothing steps gives provable superconvergence for SPD problems with a simple algebraic procedure.

read the letter

The key point is that embedding the current finite element solution in an enriched space and running a few smoothing iterations produces superconvergence for symmetric positive definite problems. The authors prove this for additive and multiplicative smoothers and show numerical results for Poisson, Maxwell, biharmonic, and Helmholtz. They do a good job keeping the whole procedure algebraic and simple to code. This matters when you already have a working finite element solver and do not want to rewrite the discretization for higher accuracy. The method applies to high-order and three-dimensional cases without extra mesh work. The proof is stated clearly for the SPD setting, and the tests span several equation types. The soft spot is the split between proved and observed behavior. Only the SPD case has analysis; the other examples depend on computation alone. Without comparisons to recovery-based or other postprocessors it is hard to know the practical edge this method has. The choice of enriched space is central but its influence on the constants is not examined much. This is for people who implement or use finite element methods in applications and want an easy postprocessing step to reduce error. A reader looking for reproducible algebraic improvements will get something concrete from it. The paper has enough structure with the proof and the experiments to merit a serious referee. I would send it to peer review.

Referee Report

1 major / 2 minor

Summary. The paper develops a smoothing-based postprocessing method for superconvergence in finite element methods. It applies a few smoothing iterations (damped Jacobi, Gauss-Seidel, or conjugate gradient) starting from the finite element solution embedded in an enriched finite element space. For symmetric and positive-definite problems, the authors prove superconvergence of the smoothed solutions under additive and multiplicative smoothers. The method is algebraic and easy to implement for high-order and three-dimensional discretizations. Effectiveness is demonstrated through numerical experiments on the Poisson, Maxwell, biharmonic, and Helmholtz equations.

Significance. Should the proof be correct, this approach provides an algebraic and straightforward way to achieve superconvergence in FEM, which is significant because it avoids the need for mesh refinement or complex postprocessing operators. It is particularly promising for high-order elements and 3D problems, and the separation of the SPD proof from experiments on other equations allows for clear scoping of the theoretical result. The method could be adopted in practice for improving solution accuracy with minimal additional cost.

major comments (1)

§3 (theoretical analysis): The central claim that a fixed number of additive or multiplicative smoothing steps applied to the embedded finite-element solution yields superconvergence for SPD problems is load-bearing, but the manuscript provides no detailed derivation, key lemmas on the approximation properties of the enriched space, contraction estimates for the smoothers, or explicit superconvergence rates. Without these, the proof cannot be verified.

minor comments (2)

The abstract and introduction would benefit from explicitly stating the superconvergence rates proved for SPD problems and observed in the experiments.
Numerical experiments lack tables of error norms, observed orders, and direct comparisons to the unsmoothed solution, which would strengthen the demonstration of effectiveness.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive assessment of its potential significance. We address the major comment below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

Referee: §3 (theoretical analysis): The central claim that a fixed number of additive or multiplicative smoothing steps applied to the embedded finite-element solution yields superconvergence for SPD problems is load-bearing, but the manuscript provides no detailed derivation, key lemmas on the approximation properties of the enriched space, contraction estimates for the smoothers, or explicit superconvergence rates. Without these, the proof cannot be verified.

Authors: We agree that the theoretical analysis requires additional explicit details for full verifiability. In the revised manuscript we will insert the missing key lemmas on the approximation properties of the enriched space, derive the contraction estimates for the additive and multiplicative smoothers (including the dependence on the damping parameter and the number of iterations), and obtain the explicit superconvergence rates in terms of the mesh size h. These additions will be placed in Section 3 while keeping the overall algebraic character of the method unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper claims an algebraic postprocessing procedure whose superconvergence is proved for symmetric positive-definite problems and demonstrated numerically on other equations. The abstract and description separate the proof from the numerical tests and from any fitting or self-referential definition of the output quantities. No load-bearing step reduces by construction to a fitted parameter, a self-citation chain, or a renamed input; the central result is presented as an independent theorem under stated coercivity assumptions. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract does not introduce new free parameters or invented entities; the central claim relies on standard finite element theory and the domain assumption of symmetric positive definite operators for the proof.

axioms (1)

domain assumption The problems are symmetric and positive definite
Explicitly required for the proof of superconvergence under additive and multiplicative smoothers.

pith-pipeline@v0.9.0 · 5395 in / 1217 out tokens · 38279 ms · 2026-05-08T16:22:54.266736+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

42 extracted references · 28 canonical work pages

[1]

D. N. Arnold and F. Brezzi , Mixed and nonconforming ﬁnite element methods: implemen- tation, postprocessing and error estimates , RAIRO Mod´ el. Math. Anal. Num´ er., 19 (1985), pp. 7–32, https://doi.org/10.1051/m2an/1985190100071

work page doi:10.1051/m2an/1985190100071 1985
[2]

D. N. Arnold, R. S. F alk, and R. Winther , Multigrid in H(div) and H(curl) , Numer. Math., 85 (2000), pp. 197–217, https://doi.org/10.1007/s002110000137

work page doi:10.1007/s002110000137 2000
[3]

R. E. Bank , Hierarchical bases and the ﬁnite element method , in Acta numerica, 1996, vol. 5 of Acta Numer., Cambridge Univ. Press, Cambridge, 1996, pp. 1–43, https://doi.org/10. 1017/S0962492900002610

1996
[4]

R. E. Bank and C. C. Douglas , Sharp estimates for multigrid rates of convergence with general smoothing and acceleration , SIAM J. Numer. Anal., 22 (1985), pp. 617–633. 20 Y. LI, H. SHUI, AND L. ZIKATANOV

1985
[5]

R. E. Bank and Y. Li , Superconvergent recovery of Raviart-Thomas mixed ﬁnite el ements on triangular grids , J. Sci. Comput., 81 (2019), pp. 1882–1905, https://doi.org/10.1007/ s10915-019-01068-0

2019
[6]

R. E. Bank and H. Nguyen , hp adaptive ﬁnite elements based on derivative recovery and superconvergence, Comput. Vis. Sci., 14 (2011), pp. 287–299, https://doi.org/10.1007/ s00791-012-0179-7

2011
[7]

R. E. Bank and R. K. Smith , A posteriori error estimates based on hierarchical bases , SIAM J. Numer. Anal., 30 (1993), pp. 921–935, https://doi.org/10.1137/0730048

work page doi:10.1137/0730048 1993
[8]

R. E. Bank and J. Xu , Asymptotically exact a posteriori error estimators. I. Gri ds with superconvergence, SIAM J. Numer. Anal., 41 (2003), pp. 2294–2312, https://doi.org/10. 1137/S003614290139874X

2003
[9]

R. E. Bank and J. Xu , Asymptotically exact a posteriori error estimators. II. Ge neral un- structured grids , SIAM J. Numer. Anal., 41 (2003), pp. 2313–2332, https://doi.org/10. 1137/S0036142901398751

2003
[10]

R. E. Bank, J. Xu, and B. Zheng , Superconvergent derivative recovery for Lagrange triangular elements of degree p on unstructured grids , SIAM J. Numer. Anal., 45 (2007), pp. 2032– 2046, https://doi.org/10.1137/060675174

work page doi:10.1137/060675174 2007
[11]

S. C. Brenner and L. R. Scott , The mathematical theory of ﬁnite element methods , vol. 35 of Texts in Applied Mathematics, 15, Springer, New York, 3 ed ., 2008

2008
[12]

S. C. Brenner and L.-Y. Sung , C0 interior penalty methods for fourth order elliptic boundar y value problems on polygonal domains , J. Sci. Comput., 22/23 (2005), pp. 83–118, https:// doi.org/10.1007/s10915-004-4135-7

work page doi:10.1007/s10915-004-4135-7 2005
[13]

Chen and Y

X. Chen and Y. Li , Superconvergent pseudostress-velocity ﬁnite element met hods for the Oseen equations, J. Sci. Comput., 92 (2022), pp. Paper No. 17, 27, https://doi.org/10.1007/ s10915-022-01856-1

2022
[14]

Cockburn, G

B. Cockburn, G. Fu, and F. J. Sayas , Superconvergence by M -decompositions. Part I: General theory for HDG methods for diﬀusion , Math. Comp., 86 (2017), pp. 1609–1641, https://doi.org/10.1090/mcom/3140

work page doi:10.1090/mcom/3140 2017
[15]

Cockburn, J

B. Cockburn, J. Guzm ´an, and H. W ang , Superconvergent discontinuous Galerkin methods for second-order elliptic problems , Math. Comp., 78 (2009), pp. 1–24, https://doi.org/10. 1090/S0025-5718-08-02146-7

2009
[16]

Y. Du, H. Wu, and Z. Zhang , Superconvergence analysis of linear FEM based on polynomia l preserving recovery for Helmholtz equation with high wave n umber, J. Comput. Appl. Math., 372 (2020), pp. 112731, 16, https://doi.org/10.1016/j.cam.2020.112731

work page doi:10.1016/j.cam.2020.112731 2020
[17]

URLhttps://doi.org/10

C. Geuzaine and J.-F. c. Remacle , Gmsh: A 3-D ﬁnite element mesh generator with built- in pre- and post-processing facilities , Internat. J. Numer. Methods Engrg., 79 (2009), pp. 1309–1331, https://doi.org/10.1002/nme.2579

work page doi:10.1002/nme.2579 2009
[18]

W.-m. He, R. Lin, and Z. Zhang , Ultraconvergence of ﬁnite element method by Richardson extrapolation for elliptic problems with constant coeﬃcie nts, SIAM J. Numer. Anal., 54 (2016), pp. 2302–2322, https://doi.org/10.1137/15M1031710

work page doi:10.1137/15m1031710 2016
[19]

X. He, Y. Chen, H. Ji, and H. W ang , Superconvergence of unﬁtted Rannacher-Turek noncon- forming element for elliptic interface problems , Appl. Numer. Math., 203 (2024), pp. 32–51, https://doi.org/10.1016/j.apnum.2024.05.016

work page doi:10.1016/j.apnum.2024.05.016 2024
[20]

Hiptmair , Multigrid method for Maxwell’s equations , SIAM J

R. Hiptmair , Multigrid method for Maxwell’s equations , SIAM J. Numer. Anal., 36 (1999), pp. 204–225, https://doi.org/10.1137/S0036142997326203

work page doi:10.1137/s0036142997326203 1999
[21]

Hiptmair and J

R. Hiptmair and J. Xu , Nodal auxiliary space preconditioning in H(curl) and H(div) spaces, SIAM J. Numer. Anal., 45 (2007), pp. 2483–2509, https://doi.org/10.1137/060660588

work page doi:10.1137/060660588 2007
[22]

J. Hu, L. Ma, and R. Ma , Optimal superconvergence analysis for the Crouzeix-Ravia rt and the Morley elements , Advances in Computational Mathematics, 47 (2021), https://doi. org/10.1007/s10444-021-09874-7

work page doi:10.1007/s10444-021-09874-7 2021
[23]

Huang, J

Y. Huang, J. Li, C. Wu, and W. Yang , Superconvergence analysis for linear tetrahe- dral edge elements , J. Sci. Comput., 62 (2015), pp. 122–145, https://doi.org/10.1007/ s10915-014-9848-7

2015
[24]

Kim , Guaranteed and asymptotically exact a posteriori error est imator for lowest-order Raviart-Thomas mixed ﬁnite element method , Appl

K.-Y. Kim , Guaranteed and asymptotically exact a posteriori error est imator for lowest-order Raviart-Thomas mixed ﬁnite element method , Appl. Numer. Math., 165 (2021), pp. 357– 375, https://doi.org/10.1016/j.apnum.2021.03.002

work page doi:10.1016/j.apnum.2021.03.002 2021
[25]

Li , Recovery-based a posteriori error analysis for plate bendi ng problems , J

Y. Li , Recovery-based a posteriori error analysis for plate bendi ng problems , J. Sci. Comput., 88 (2021), pp. Paper No. 77, 26, https://doi.org/10.1007/s10915-021-01595-9

work page doi:10.1007/s10915-021-01595-9 2021
[26]

Li , Superconvergent ﬂux recovery of the Rannacher-Turek nonco nforming element , J

Y. Li , Superconvergent ﬂux recovery of the Rannacher-Turek nonco nforming element , J. Sci. Comput., 87 (2021), pp. Paper No. 32, 19, https://doi.org/10.1007/s10915-021-01445-8

work page doi:10.1007/s10915-021-01445-8 2021
[27]

Li and H

Y. Li and H. Shui , Smoother-type a posteriori error estimates for ﬁnite eleme nt methods , Comput. Methods Appl. Mech. Engrg., 453 (2026), pp. Paper No . 118847, 21, https://doi. SUPERCONVERGENCE IN FEM BY SMOOTHING 21 org/10.1016/j.cma.2026.118847

work page doi:10.1016/j.cma.2026.118847 2026
[28]

Computers & Mathematics with Applications132, 48– 62 (2023) https://doi.org/10.1016/j.camwa

Y. Li and L. Zikatanov , A posteriori error estimates of ﬁnite element methods by pre condi- tioning, Comput. Math. Appl., 91 (2021), pp. 192–201, https://doi.org/10.1016/j.camwa. 2020.08.001

work page doi:10.1016/j.camwa 2021
[29]

Nodal auxiliary a posteri ori error estimates

Y. Li and L. Zikatanov , Nodal auxiliary a posteriori error estimates , Math. Comp., (2025), https://doi.org/10.1090/mcom/4141

work page doi:10.1090/mcom/4141 2025
[30]

Y. Li, L. Zikatanov, and C. Zuo , A reduced conjugate gradient basis method for fractional di f- fusion, SIAM J. Sci. Comput., (2024), pp. S68–S87, https://doi.org/10.1137/23M1575913

work page doi:10.1137/23m1575913 2024
[31]

Y. Li, L. Zikatanov, and C. Zuo , Reduced Krylov basis methods for parametric partial diﬀer- ential equations , SIAM J. Numer. Anal., 63 (2025), pp. 976–999, https://doi.org/10.1137/ 24M1661236

2025
[32]

Li , Global superconvergence of the lowest-order mixed ﬁnite el ement on mildly struc- tured meshes , SIAM J

Y.-W. Li , Global superconvergence of the lowest-order mixed ﬁnite el ement on mildly struc- tured meshes , SIAM J. Numer. Anal., 56 (2018), pp. 792–815, https://doi.org/10.1137/ 17M112587X

2018
[33]

Lin and Z

R. Lin and Z. Zhang , Natural superconvergence points in three-dimensional ﬁni te elements , SIAM J. Numer. Anal., 46 (2008), pp. 1281–1297, https://doi.org/10.1137/070681168

work page doi:10.1137/070681168 2008
[34]

Mulita, S

O. Mulita, S. Giani, and L. Heltai , Quasi-optimal mesh sequence construction through smoothed adaptive ﬁnite element methods , SIAM J. Sci. Comput., 43 (2021), pp. A2211– A2241, https://doi.org/10.1137/19M1262097

work page doi:10.1137/19m1262097 2021
[35]

L. B. W ahlbin, Superconvergence in Galerkin ﬁnite element methods , Lecture Notes in Math- ematics, 1605, Springer-Verlag, Berlin, 1995

1995
[36]

Iterative methods by space decomposition and subspace correction

J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev., 34 (1992), pp. 581–613, https://doi.org/10.1137/1034116

work page doi:10.1137/1034116 1992
[37]

Xu and L

J. Xu and L. Zikatanov , The method of alternating projections and the method of subs pace corrections in Hilbert space , J. Amer. Math. Soc., 15 (2002), pp. 573–597, https://doi.org/ 10.1090/S0894-0347-02-00398-3

work page doi:10.1090/s0894-0347-02-00398-3 2002
[38]

Ye and S

X. Ye and S. Zhang , Order two superconvergence of the CDG ﬁnite elements on tria ngular and tetrahedral meshes , CSIAM Trans. Appl. Math., 4 (2023), pp. 256–274, https://doi. org/10.4208/csiam-am.SO-2021-0051

work page doi:10.4208/csiam-am.so-2021-0051 2023
[39]

Zhang, Y

Y. Zhang, Y. Chen, Y. Huang, and N. Yi , Superconvergent cluster recovery method for the Crouzeix-Raviart element , Numer. Math. Theory Methods Appl., 14 (2021), pp. 508–526, https://doi.org/10.4208/nmtma.oa-2020-0117

work page doi:10.4208/nmtma.oa-2020-0117 2021
[40]

Zhang and A

Z. Zhang and A. Naga , A new ﬁnite element gradient recovery method: superconverg ence property, SIAM J. Sci. Comput., 26 (2005), pp. 1192–1213, https://doi.org/10.1137/ S1064827503402837

2005
[41]

O. C. Zienkiewicz and J. Z. Zhu , The superconvergent patch recovery and a posteriori er- ror estimates. I. The recovery technique , Internat. J. Numer. Methods Engrg., 33 (1992), pp. 1331–1364, https://doi.org/10.1002/nme.1620330702

work page doi:10.1002/nme.1620330702 1992
[42]

L. T. Zikatanov , Two-sided bounds on the convergence rate of two-level metho ds, Numer. Linear Algebra Appl., 15 (2008), pp. 439–454, https://doi.org/10.1002/nla.556

work page doi:10.1002/nla.556 2008