Pith. sign in

REVIEW 4 major objections 7 minor 1 cited by

Gevrey class regularity for steady-state incompressible Navier-Stokes equations in parametric domains and related models

T0 review · 4 major / 7 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For steady Navier-Stokes with random parameters, velocity and pressure inherit Gevrey-delta regularity from the coefficients and data.

desk verdict First genuine Gevrey-δ regularity for the velocity–pressure pair of stationary Navier–Stokes in parametric domains; the induction closes, but the uniform inf-sup assumption is load-bearing and unverified in the domain examples. read the letter →

arxiv 2504.13753 v1 pith:7XFH47O5 submitted 2025-04-18 math.NA cs.NA

Alexey Chernov , Tung Le This is my paper · ORCID
classification math.NAcs.NA MSC 35Q3065C3065D3065D3265N30
keywords Navier-Stokesequationparametricregularityanalysisquasi-MonteCarlomethodsuncertaintyquantificationGevreydomainssaddle-pointproblemsalternative-to-factorialtechnique
topics Navier-Stokes Regularity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that the solution pair of the steady incompressible Navier-Stokes equations, stated in mixed form with coefficients, domain, force, and source term all depending on a high-dimensional parameter $y$, is as smooth in $y$ as the given data are, in the Gevrey sense. Under a small-data condition and a uniform inf-sup assumption, every mixed derivative of velocity and pressure satisfies a bound with $(|\nu|!)^\delta$ growth, explicit constants, and a common radius sequence. Explicit constants matter because they feed directly into Gauss-Legendre and quasi-Monte Carlo convergence rates for expected quantities such as mean velocity norms or pressure norms. Analytic regularity is the special case $\delta=1$, and the numerical experiments confirm the predicted convergence rates.

What carries the argument

The engine is the alternative-to-factorial technique: instead of allowing $|\nu|!$ to accumulate through Leibniz-type product rules, all estimates are written with falling factorials $(\tfrac12)_n$, whose binomial convolutions are summable by the explicit identities (7)--(12) in Lemma 1. Around that, the proof uses the linearized form $t_y(u,w,v)=a(w,v;A(y))+m(u,w,v;M(y))+m(w,u,v;M(y))$, whose uniform coercivity $\alpha-m\gamma$ is guaranteed by the small-data assumption, and the inf-sup condition (30), which makes $B_y$ and $B_y^\top$ isomorphisms with norms controlled by $\beta$. For the domain-transformation part, the multivariate Faà di Bruno formula together with the exact combinatorial identity of Lemma 4 converts composite pullbacks of $f$ and $g$ into Gevrey-$\delta$ bounds with explicit constants.

What would settle it

Compute the inf-sup constant $\beta(y)$ in Assumption 2 for the domain-transformation family of Example 1 by a standard numerical approximation method for inf-sup constants; if $\beta(y)\to 0$ while the matrices and data remain uniformly Gevrey-$\delta$, then no uniform $\beta$ exists and the uniform regularity statement of Theorem 2 is vacuous for that family.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2: if the matrices $A(y),B(y),M(y)$, the force $f(y)$, and the source mass term $g(y)$ in (39) satisfy Assumptions 1--4 for some $\delta\ge 1$, then the solution pair $(u,p)$ is of class Gevrey-$\delta$ in $y$. Concretely, there exist constants $\tilde C_u, \tilde C_p>0$ and a positive sequence $\tilde R$ such that $\|\partial^\nu u\|_H \le \tilde C_u(|\nu|!)^\delta/\tilde R^\nu$ and $\|\partial^\nu p\|_L \le \tilde C_p(|\nu|!)^\delta/\tilde R^\nu$ for every multi-index $\nu$. The proof is an induction on derivative order: the highest-order velocity derivative is split into a divergence-free part and a potential part, the divergence-free part is controlled through the inf-sup isomorphism, the potential part through coercivity of the linearized form, and the pressure derivative is then read off from the momentum equation. All constants are assembled explicitly in (54), so the Gevrey radius and prefactors are known in terms of the problem data. Section 5 shows that the plain pullback of a Gevrey-$\delta$ parametric domain transformation makes the matrices and pulled-back data satisfy the same assumptions, so the theorem covers randomly perturbed Lipschitz domains.

Load-bearing premise

The load-bearing premise is Assumption 2: a single fixed constant $\beta$ must bound the pressure-velocity coupling from below for every parameter value at once; if that coupling weakens anywhere in the parameter range, the uniform Gevrey constants in the conclusion blow up, and for parametric domains such a uniform bound is assumed rather than proved for the transformation examples.

Editorial extensions

If this is right

  • For scalar parameters, Gauss-Legendre quadrature of solution functionals converges like $C\exp(-r n^{1/\delta})$, and the experiments confirm this for analytic ($\delta=1$) and Gevrey-$\tfrac32$ perturbations.
  • For high-dimensional parameters generated by $\ell_p$-summable series, randomly shifted rank-1 lattice QMC rules achieve errors close to $O(n^{-1})$ for both analytic and Gevrey-$\tfrac32$ inputs whenever $p<2/(3\delta)$.
  • The QMC convergence-rate bound is dimension-independent, so the relative error does not grow with the truncation dimension $s$ in the tested setting.
  • Parametric domain uncertainty falls under the same theorem: transforming back to a nominal domain preserves Gevrey-$\delta$ regularity of the coefficients and data, so randomly perturbed domains inherit the same quadrature consequences.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because all constants in the derivative bounds are explicit, a natural next step is to turn the formulas into a priori error estimators for choosing QMC sample sizes; the paper stops at convergence-rate predictions rather than computable bounds.
  • The uniform inf-sup assumption is the most fragile input for parametric domains; replacing it with a parameter-dependent $\beta(y)$ and weighted Gevrey constants, or verifying uniformity numerically for each transformation family, would broaden the theorem's applicability.
  • The falling-factorial induction appears to transfer to other saddle-point systems such as parametrized Stokes or Oseen equations, where the same coercivity and inf-sup isomorphism structure is present.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 7 minor

Summary. The paper investigates parametric regularity of the steady incompressible Navier-Stokes equations in the mixed variational form (1), where the coefficients A(y), B(y), M(y) and the data f(y), g(y) belong to a Gevrey class of index δ≥1. The main results (Theorems 1 and 2) state that, under small-data (Assumption 3), uniform coercivity and inf-sup (Assumption 2), and Gevrey-δ data (Assumption 4), the solution pair (u,p) satisfies explicit Gevrey-δ derivative bounds with constants C_u, C_p, and ρ. The proof employs the alternative-to-factorial technique: an induction over the derivative order with a decomposition of ∂^{ν+e}u into V and V⊥ components, using Lemmas 13–15 and the combinatorial estimates of Lemma 16. Section 5 aims to verify Assumption 4 for the plain pullback of parametric domain perturbations (Theorems 3 and 4), and Section 6 provides numerical experiments with Gauss-Legendre quadrature and Quasi-Monte Carlo methods that reproduce the predicted convergence rates.

Significance. The result is significant because explicit Gevrey-δ regularity of the full solution pair (velocity and pressure) for the saddle-point Navier-Stokes system is new and directly supports convergence analyses of high-dimensional quadrature methods; the analytic case δ=1 is recovered as a special case. The paper also provides explicit constants, which is valuable for practitioners. The numerical experiments are careful and reproduce the predicted rates of Gauss-Legendre and QMC convergence. The proof is structurally sound: the coercivity of the linearized form, the inf-sup isomorphism, and the combinatorial estimates close the induction, provided the typos in the displayed constants are corrected.

major comments (4)
  1. [§5 (Theorems 3–4), §6, and Assumption 2 (Eq. (30))] The uniform inf-sup condition, Assumption 2 (Eq. (30)), is never verified for the parametric-domain transformations, yet Theorem 2 and the bounds (52)–(53) depend on a single β>0 for all y∈U through Lemmas 13 (Eq. (55)) and 15 (Eq. (63)), which divide by β. If inf_{y∈U} β(y)=0 for the transforms in Example 1 or in the numerical examples of Section 6, the Gevrey constants C̃_u, C̃_p, R̃ in Theorem 2 are not finite. I request either a proof of a uniform lower bound for β(y) for these families of transformations, or an explicit caveat that the parametric-domain application requires a separate verification of Assumption 2.
  2. [§4, Eq. (54)] The displayed formula for ρ_p omits the term 2C_p b that appears in the numerator of the bound (74) used in the pressure induction step. With the printed definition of ρ_p, the inequality '≤ C_p ρ_p ρ^{|ν|-1}...' in the proof of Theorem 1 does not close. The correct numerator should include +2C_p b, matching (74); please verify and correct the constant.
  3. [§4, Lemma 16 (Eqs. (66)–(72)) and Theorem 1 proof (Eqs. (73)–(75))] The factorial exponent is printed as 1−δ everywhere in these displayed inequalities, but the detailed proof of Lemma 16 and the induction requirement use δ−1. As printed, these inequalities are false for δ>1. In addition, the induction hypothesis (64) uses [1/2]^{|ν|} where [1/2]^{|η|} is required. These are systematic typographical errors and must be corrected throughout.
  4. [§5, Theorem 3] The statement B(y)=M(y)=dT^{−⊤} det(dT) is inconsistent with Example 1 and with the standard pullback of the divergence and convection terms, which yield B(y)=M(y)=dT^{−1} det(dT). The regularity proof is unaffected because it treats dT^{−1} and dT^{−⊤} identically, but the displayed formulation of the theorem should be corrected.
minor comments (7)
  1. [§3, Lemma 7] The closedness of R(B_y^T) is justified by the boundedness of B_y via (33), but boundedness alone does not imply closed range; the proof should invoke Assumption 2.
  2. [§4, Lemma 15] The statement refers to A2(ν), but Lemma 14 defines A(ν); harmonize the notation.
  3. [§4, Lemma 16 (hypothesis (65))] The pressure derivative is measured in the L-norm, not the H-norm as printed.
  4. [§6.2, after Eq. (96)] The statement that the relative error is independent of dimensions is inconsistent with the s-dependence in Eq. (95); rephrase to say that the observed rates are nearly dimension-independent for the tested examples.
  5. [§6.1] The number '16.641 degrees of freedom' should be written '16,641' if the thousands separator is intended.
  6. [Assumption 4] Assumption 4 restricts to s<∞ while Definition 1 and Theorem 2 allow countably many parameters; clarify that the proof applies to finite-dimensional truncations with R independent of s.
  7. [Abstract] The abstract contains the typo 'parameteric'; it should read 'parametric'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the Gevrey regularity of the solution is derived from explicit assumptions on the data with explicit constants; only auxiliary self-citations appear.

full rationale

The main derivation is self-contained and non-circular. Theorem 2 is proved by induction (Theorem 1) under Assumptions 1–4, with all constants C_u, C_p, sigma_u, sigma_p, rho_u, rho_p defined explicitly in (54) and the Gevrey bounds for the solution obtained from the assumed Gevrey bounds for A,B,M,f,g. The conclusion is not an input by construction: the solution's regularity is a transfer result, and the proof isolates the highest-order derivative via Lemmas 13–15 and the combinatorial bounds in Lemma 16. Section 5 verifies Assumption 4 for the plain pullback using Lemma 2 and the multivariate Faà di Bruno formula, which are independent of the Navier-Stokes theorem. The paper does cite the authors' own prior works for auxiliary combinatorial inequalities (Lemma 1, from [10,11]) and for the QMC convergence rate for Gevrey integrands ([10, Lemma 7.4]), but these are separate published results and are not the claim being established; they do not reduce the main theorem to a restatement of its inputs. No fitted parameter is renamed as a prediction: the numerical experiments compare observed quadrature slopes with the fixed theoretical rates (87), (88), and (95). Two non-circular concerns should be recorded: Assumption 2's y-uniform inf-sup condition (30) is assumed rather than verified for the parametric-domain examples, so the uniformity of the constants in Theorem 2 for those examples rests on an unproved assumption; and Lemma 16 displays the factorial exponent (|nu+e|!)^{1-delta} while the proof line and Assumption 4 give (|nu+e|!)^{delta-1}, an apparent sign typo. Neither issue makes the derivation circular.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central theorem assumes a small-data regime and uniform stability; no parameters are fitted to data. Section 5 adds geometric Gevrey assumptions on the domain map. The only imported mathematical tools are standard Sobolev embedding, the Faà di Bruno formula, and elementary falling-factorial inequalities cited from the authors' prior work.

assumptions (8)
  • domain assumption Uniform inf-sup condition: there exists beta > 0 independent of y such that inf over nonzero q in L of sup over nonzero v in H of b(v,q;B(y))/(||v||_H ||q||_L) >= beta, Assumption 2, Eq. (30).
    Used repeatedly to bound the pressure derivatives and the w_perp component (Lemmas 9, 13, 15); for parametric domains this uniformity is not automatic and may fail for large deformations.
  • domain assumption Small-data condition f/alpha + g/beta < alpha/m, Assumption 3.
    Guarantees well-posedness in the ball B(0,gamma) and strict coercivity alpha_tilde = alpha - m*gamma > 0 of the linearized form t_y (Lemma 12); the entire induction in Theorem 1 uses gamma < alpha/m.
  • domain assumption Coefficient Gevrey-delta input bounds on A, B, M, f, and g, Assumption 4, Eq. (51).
    The central smoothness assumption on the data; the theorem states that this regularity is inherited by the solution.
  • domain assumption Uniform bounded invertibility of the domain transformation T_y and its differential, and Gevrey-delta of T_y, Assumption 5.
    Needed in Section 5 to transfer Gevrey regularity to A, B, M and to compose f, g with the pullback.
  • domain assumption Gevrey-delta regularity in x of the physical data f and g on the hold-all domain, Assumption 6.
    Required for the Faà di Bruno argument in Lemma 17 bounding the pullback of the force and mass terms.
  • standard math Sobolev embedding H^1_0(D) into L^4(D) with constant C_4, used in Assumption 1 and Lemma 5.
    Boundedness of the trilinear form m in (34) relies on it.
  • standard math The combinatorial falling-factorial inequalities of Lemma 1, with proofs in [10, Section 7] and [11, Section 2].
    Elementary but non-obvious identities imported from the authors' prior work; the regularity induction depends on them.
  • standard math The multivariate Faà di Bruno formula of Constantine and Savits, [16, Theorem 2.1].
    Used in Lemma 17 for the pullback of the force term.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Gevrey class regularity for steady-state incompressible Navier-Stokes equations in parametric domains and related models." pith.science (2026). https://pith.science/paper/7XFH47O5

@misc{pith2026250413753,
  author       = {Pith},
  title        = {Pith review of: Gevrey class regularity for steady-state incompressible Navier-Stokes equations in parametric domains and related models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7XFH47O5}},
  note         = {Machine review of arXiv:2504.13753}
}
read the original abstract

We investigate parameteric Navier-Stokes equations for a viscous, incompressible flow in bounded domains. The coefficients of the equations are perturbed by high-dimensional random parameters, this fits in particular for modelling flows in domains with uncertain perturbations. Our focus is on deriving bounds for arbitrary high-order derivatives of the pressure and the velocity fields with respect to the random parameters in the context of incompressible Navier-Stokes equation under a small-data assumption. To achieve this, we analyze mixed and saddle-point problems and employ the alternative-to-factorial technique to establish generalized Gevrey-class regularity for the solution pair. Thereby the analytic regularity follows as a special case. In the numerical experiments, we validate and illustrate our theoretical findings using Gauss-Legendre quadrature and Quasi-Monte Carlo methods.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Quasi-Monte Carlo methods for uncertainty quantification of tumor growth modeled by a parametric semi-linear parabolic reaction-diffusion equation

    math.NA 2025-09 accept novelty 8.0 of 10

    For a semilinear parabolic tumor-growth PDE with uniform random coefficients, quasi-Monte Carlo achieves near-linear error convergence O(N^{-1}) for output expectations, beating Monte Carlo's O(N^{-1/2}), with proof a...

Reference graph

Works this paper leans on

36 extracted references · 28 canonical work pages · cited by 1 Pith paper

  1. [1]

    Sperone, G.: Steady-state Navier-Stokes flow in an obstructed pipe under mixed boundary conditions and with a prescribed transversal flux rate. Calc. Var. Partial Differential Equations 62(9), 236–57 (2023) https://doi.org/10.1007/ s00526-023-02578-4 40

    work page 2023

  2. [2]

    Springer series in computational mathematics

    John, V.: Finite Element Methods for Incompressible Flow Problems, 1st edn. Springer series in computational mathematics. Springer, Cham, Switzerland (2016)

    work page 2016

  3. [3]

    Research report, Université de lorraine (April 2013)

    Plotnikov, P.I., Sokolowski, J.: Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations. Research report, Université de lorraine (April 2013). https://inria.hal.science/hal-00807319

    work page 2013

  4. [4]

    Nonlinear Anal

    Constantin, A., Johnson, R.S.: On the dynamics of the near-surface currents in the Arctic Ocean. Nonlinear Anal. Real World Appl.73, 103894–43 (2023) https://doi.org/10.1016/j.nonrwa.2023.103894

    arXiv 2023

  5. [5]

    Deep Sea Research Part II: Topical Studies in Oceanography160, 32–40 (2019) https://doi.org/10.1016/j.dsr2.2018.12.007

    Constantin, A., Johnson, R.S.: Large-scale oceanic currents as shallow-water asymptotic solutions of the Navier-Stokes equation in rotating spherical coordi- nates. Deep Sea Research Part II: Topical Studies in Oceanography160, 32–40 (2019) https://doi.org/10.1016/j.dsr2.2018.12.007

    work page doi:10.1016/j.dsr2.2018.12.007 2019

  6. [6]

    Mathematics of Cli- mate and Weather Forecasting 5(1), 65–106 (2019) https://doi.org/10.1515/ mcwf-2019-0005

    Chertock, A., Kurganov, A., Lukáčová-Medvid’ová, M., Spichtinger, P., Wiebe, B.: Stochastic Galerkin method for cloud simulation. Mathematics of Cli- mate and Weather Forecasting 5(1), 65–106 (2019) https://doi.org/10.1515/ mcwf-2019-0005

    work page 2019

  7. [7]

    part ii: A fully random navier-stokes-cloud model

    Chertock, A., Kurganov, A., Lukáčová-Medvid’ová, M., Spichtinger, P., Wiebe, B.: Stochastic Galerkin method for cloud simulation. part ii: A fully random navier-stokes-cloud model. Journal of Computational Physics479, 111987 (2023) https://doi.org/10.1016/j.jcp.2023.111987

    arXiv 2023

  8. [8]

    SIAM Journal on Mathematical Analysis 50(2), 1720–1752 (2018) https://doi.org/10.1137/16m1099406

    Cohen, A., Schwab, C., Zech, J.: Shape holomorphy of the stationary Navier– Stokes equations. SIAM Journal on Mathematical Analysis 50(2), 1720–1752 (2018) https://doi.org/10.1137/16m1099406

    work page doi:10.1137/16m1099406 2018

Show all 36 references
  1. [9]

    https://arxiv.org/abs/2210.15572

    Ko, S., Li, G., Yu, Y.: Quasi-Monte Carlo finite element approximation of the Navier-Stokes equations with initial data modeled by log-normal random fields (2025). https://arxiv.org/abs/2210.15572

    2025 arXiv

  2. [10]

    SIAM Journal on Numerical Analysis 62(4), 1874–1900 (2024) https://doi.org/10.1137/23M1596296

    Chernov, A., Lê, T.: Analytic and Gevrey class regularity for parametric ellip- tic eigenvalue problems and applications. SIAM Journal on Numerical Analysis 62(4), 1874–1900 (2024) https://doi.org/10.1137/23M1596296

    2024 doi

  3. [11]

    Computers & Mathematics with Applications164, 116–130 (2024) https://doi

    Chernov, A., Lê, T.: Analytic and Gevrey class regularity for parametric semi- linear reaction-diffusion problems and applications in uncertainty quantification. Computers & Mathematics with Applications164, 116–130 (2024) https://doi. org/10.1016/j.camwa.2024.04.007

    2024 doi

  4. [12]

    Mathemati- cal Models and Methods in Applied Sciences (2024) https://doi.org/10.1142/ 41 s0218202524500179

    Harbrecht, H., Schmidlin, M., Schwab, C.: The Gevrey class implicit mapping theorem with application to UQ of semilinear elliptic PDEs. Mathemati- cal Models and Methods in Applied Sciences (2024) https://doi.org/10.1142/ 41 s0218202524500179

    2024

  5. [13]

    https://arxiv.org/abs/2502.12345

    Djurdjevac, A., Kaarnioja, V., Schillings, C., Zepernick, A.-A.: Uncertainty quan- tification for stationary and time-dependent PDEs subject to Gevrey regular random domain deformations (2025). https://arxiv.org/abs/2502.12345

    2025 arXiv

  6. [14]

    An- nales de l’institut Fourier 17(1), 295–323 (1967) https://doi.org/10.5802/aif

    Monvel, L.B.D., Krée, P.: Pseudo-differential operators and Gevrey classes. An- nales de l’institut Fourier 17(1), 295–323 (1967) https://doi.org/10.5802/aif. 258

    1967 doi

  7. [15]

    Foundations of Computational Mathematics 10(6), 615–646 (2010) https://doi.org/10.1007/s10208-010-9072-2

    Cohen, A., DeVore, R., Schwab, C.: Convergence rates of best N-term Galerkin approximations for a class of elliptic sPDEs. Foundations of Computational Mathematics 10(6), 615–646 (2010) https://doi.org/10.1007/s10208-010-9072-2

    2010 doi

  8. [16]

    Transactions of the American Mathematical Society348(2), 503–520 (1996)

    Constantine, G.M., Savits, T.H.: A multivariate Faa di Bruno formula with ap- plications. Transactions of the American Mathematical Society348(2), 503–520 (1996)

    1996

  9. [17]

    Birkhäuser Advanced Texts Basler Lehrbücher

    Krantz, S.G., Parks, H.: A Primer of Real Analytic Functions. Birkhäuser Advanced Texts Basler Lehrbücher. Springer, New York, NY (2012)

    2012

  10. [18]

    Numerische Mathematik 142(4), 863–915 (2019) https://doi.org/10

    Gilbert, A.D., Graham, I.G., Kuo, F.Y., Scheichl, R., Sloan, I.H.: Analysis of quasi-Monte Carlo methods for elliptic eigenvalue problems with stochastic co- efficients. Numerische Mathematik 142(4), 863–915 (2019) https://doi.org/10. 1007/s00211-019-01046-6

    2019

  11. [19]

    Part I: Smooth domains

    Costabel, M., Dauge, M., Nicaise, S.: Corner Singularities and Analytic Regu- larity for Linear Elliptic Systems. Part I: Smooth domains. 211 pages (2010). https://hal.archives-ouvertes.fr/hal-00453934

    2010

  12. [20]

    American Mathematical Society, Providence, RI (2024)

    Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis. American Mathematical Society, Providence, RI (2024)

    2024

  13. [21]

    Springer series in computational mathematics

    Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations. Springer series in computational mathematics. Springer, Berlin, Germany (2011)

    2011

  14. [22]

    Mizuguchi, M., Tanaka, K., Sekine, K., Oishi, S.: Estimation of Sobolev embed- ding constant on a domain dividable into bounded convex domains. J. Inequal. Appl., 299–18 (2017) https://doi.org/10.1186/s13660-017-1571-0

    2017 doi

  15. [23]

    Mathematics of Computation 88(315), 73–89 (2018) https://doi.org/10

    Gallistl, D.: Rayleigh–Ritz approximation of the inf-sup constant for the diver- gence. Mathematics of Computation 88(315), 73–89 (2018) https://doi.org/10. 1090/mcom/3327

    2018

  16. [24]

    Classics in mathematics

    Yosida, K.: Functional Analysis, 6th edn. Classics in mathematics. Springer, Berlin, Germany (1995)

    1995

  17. [25]

    50(3), 1433–1455 (2012) https://doi.org/10.1137/100812574

    Chernov, A., Schwab, C.: Exponential convergence of Gauss-Jacobi quadratures 42 forsingularintegralsoversimplicesinarbitrarydimension.SIAMJ.Numer.Anal. 50(3), 1433–1455 (2012) https://doi.org/10.1137/100812574

    2012 doi

  18. [26]

    In: General Theory of Partial Differential Equations and Microlocal Analysis (Trieste, 1995)

    Chen, H., Rodino, L.: General theory of PDE and Gevrey classes. In: General Theory of Partial Differential Equations and Microlocal Analysis (Trieste, 1995). Pitman Res. Notes Math. Ser., vol. 349, pp. 6–81. Longman, Harlow, London, England (1996)

    1996

  19. [27]

    Silvester, D., Elman, H., Ramage, A.: Incompressible Flow and Iterative Solver Software (IFISS) version 3.5.http://www.cs.umd.edu/ elman/ifiss/ (2016)

    2016

  20. [28]

    SIAM Review56, 261–273 (2014)

    Elman, H., Ramage, A., Silvester, D.: IFISS: A computational laboratory for investigating incompressible flow problems. SIAM Review56, 261–273 (2014)

    2014

  21. [29]

    ACM Trans

    Elman, H., Ramage, A., Silvester, D.: Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow. ACM Trans. Math. Softw.33, 2–14 (2007)

    2007

  22. [30]

    Hansen, M., Schwab, C.: Analytic regularity and nonlinear approximation of a class of parametric semilinear elliptic PDEs. Math. Nachr. 286(8-9), 832–860 (2013) https://doi.org/10.1002/mana.201100131

    2013 doi

  23. [31]

    Cohen, A., Devore, R., Schwab, C.: Analytic regularity and polynomial approxi- mation of parametric and stochastic elliptic PDE’s. Anal. Appl. (Singap.)9(1), 11–47 (2011) https://doi.org/10.1142/S0219530511001728

    2011 doi

  24. [32]

    ANZIAM J

    Kuo, F.Y., Schwab, C., Sloan, I.H.: Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) set- ting and beyond. ANZIAM J. 53(1), 1–37 (2011) https://doi.org/10.1017/ S1446181112000077

    2011

  25. [33]

    SIAM Journal on Numerical Analysis50(6), 3351–3374 (2012)

    Kuo, F.Y., Schwab, C., Sloan, I.H.: Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients. SIAM Journal on Numerical Analysis50(6), 3351–3374 (2012). Accessed 2023-09-05

    2012

  26. [34]

    Kuo, F.Y., Nuyens, D.: Application of quasi-Monte Carlo methods to elliptic PDEs with random diffusion coefficients: a survey of analysis and implementa- tion. Found. Comput. Math.16(6), 1631–1696 (2016) https://doi.org/10.1007/ s10208-016-9329-5

    2016

  27. [35]

    https://people.cs.kuleuven.be/⁄tildelowdirk.nuyens/qmc4pde/

    Kuo, F.Y., Nuyens, D.: Software package QMC4PDE. https://people.cs.kuleuven.be/⁄tildelowdirk.nuyens/qmc4pde/

  28. [36]

    Cools, R., Kuo, F.Y., Nuyens, D.: Constructing embedded lattice rules for multivariate integration. SIAM J. Sci. Comput.28(6), 2162–2188 (2006) 43

    2006

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.