Pith. sign in

REVIEW 3 major objections 4 minor 50 references

Divergence-free interpolation of tangential vector fields via matrix-valued kernels

T0 review · 3 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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

Pith's one-line read This paper constructs divergence-free tangential kernels on the unit sphere and proves that a lower-order isotropic form preserves the full Sobolev regularity of the underlying scalar kernel, gaining one Sobolev order over the classical…

desk verdict The m=1 one-order gain and the multiplier-preserving inverse Laplace–Beltrami kernel are the real new results, but every multiplier identity rests on an unproved vector spherical harmonic addition formula quoted from a companion paper. read the letter →

arxiv 2608.05547 v1 pith:RLQANEBK submitted 2026-08-06 math.NA cs.NA

Zhengjie Sun , Lishuo Dong , Biao Huang This is my paper · ORCID
classification math.NAcs.NA MSC 41A0541A2543A9065D12
keywords divergence-freevectorfieldstangentialmatrix-valuedkernelszonalmultipliersuperconvergenceinterpolationonthesphere
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

This paper develops a family of matrix-valued kernels for interpolating tangential, divergence-free vector fields on the unit sphere. The central claim is that the classical potential-based construction, which applies surface differential operators to a scalar kernel, unnecessarily loses one Sobolev order of regularity: the resulting vector multipliers decay like $\ell^{-2\sigma+2}$ when the scalar zonal kernel has coefficients of order $(1+\ell(\ell+1))^{-\sigma}$. The paper shows that a lower-order isotropic construction, obtained by separating the geometric enforcement of the divergence-free constraint from the choice of scalar generator, yields multipliers of order $\ell^{-2\sigma}$ for the same scalar kernel, so the native space is the full $H^\sigma_{\rm div}(S^2)$ instead of $H^{\sigma-1}_{\rm div}(S^2)$. It further introduces an inverse Laplace--Beltrami construction whose vector multipliers equal the scalar multipliers exactly. If correct, these kernels enforce tangency and zero surface divergence at no cost in regularity, and the paper's stability and error estimates — including superconvergence for extra-smooth targets — make them directly usable for scattered-data approximation on spheres and other embedded surfaces.

What carries the argument

The load-bearing identity is the vector spherical harmonic addition formula of Lemma 2.1 (taken from the authors' earlier paper [39]), which expresses the subspace projection kernels $S_\ell(x,y)=\sum_{k=1}^{2\ell+1}y_{\ell,k}(x)y_{\ell,k}(y)^\top$ and $T_\ell(x,y)$ as linear combinations of $P_\ell''(t)$, $P_\ell'(t)$ and the geometric matrices $Q$ and $R$ defined above. This identity converts the geometric kernel (3.3) into a spectral expansion, so that positivity, native-space equivalence, stability, and convergence all become statements about the scalar Fourier difference $\widehat\varphi(\ell-1)-\widehat\varphi(\ell+1)$. The proof then applies two further quantitative tools: a filtered vector-kernel localization estimate (Lemma 5.3) bounding the kernel of a spectral cut-off in terms of $L^2(1+L\theta)^{-\nu}$, and a spherical packing bound (Lemma 5.4) that converts separation into a decay estimate for off-diagonal sums. Together these give the small-eigenvalue bound $\lambda_{\min}(A_{K_{\rm div},X})\ge C L^2\min_{1\le\ell\le L}\kappa_\ell$ with $L\sim q_X^{-1}$, from which the stability and all error estimates follow.

What would settle it

For a zonal kernel with known Fourier coefficients, e.g., the Gaussian $\varphi(t)=\exp(-\varepsilon^2(2-2t))$ restricted to $S^2$, numerically project the kernel $K_{\rm div}(\cdot,y)e$ onto vector spherical harmonics for several degrees $\ell$ and compare with the closed form in Corollary 3.2; any mismatch at a single degree would refute the main multiplier identity. Alternatively, for a smooth divergence-free target field and a scalar kernel with native space order $\sigma$, measure the interpolation error in $L^\infty$ as $h_X$ goes to zero; if the $m=1$ kernel converges at order $h_X^{\sigma-1}$ or worse instead of $h_X^{\sigma}$, the claimed one-order gain is false.

Watch

Extended reading notes

Core claim

The paper's central discovery is that the order of differentiation in the surface divergence-free kernel construction is not fixed by the constraint. On $S^2$, for every integer $m\ge 1$, the matrix kernel $K_{\rm div}(x,y)=\varphi^{(m-1)}(x\cdot y)R(x,y)+\varphi^{(m)}(x\cdot y)Q(x,y)$ is tangential and divergence-free, where $R(x,y)=tI-yx^\top$, $Q(x,y)=-(x\times y)(x\times y)^\top$, and $t=x\cdot y$. For $m=1$, the vector spherical harmonic multipliers are $\kappa_\ell = \frac{\lambda_\ell}{2\ell+1}(\widehat\varphi(\ell-1)-\widehat\varphi(\ell+1))$; when $\widehat\varphi(\ell)\asymp(1+\lambda_\ell)^{-\sigma}$ with $\lambda_\ell=\ell(\ell+1)$, this gives $\kappa_\ell\asymp(1+\lambda_\ell)^{-\sigma}$, i.e., the scalar kernel's Sobolev order is preserved. The classical $m=2$ potential-based kernel instead gives $\kappa_\ell\asymp\ell^{-2\sigma+2}$, a loss of one order. The paper also proves that the inverse Laplace--Beltrami kernel $K_{\rm div}(x,y)=\psi(t)R(x,y)+\psi'(t)Q(x,y)$ with $\psi(t)=\frac{1}{1-t^2}\int_{-1}^t(c_\varphi-\varphi(s))\,ds$ has multipliers $\kappa_\ell=\widehat\varphi(\ell)$, matching the scalar kernel exactly.

Load-bearing premise

The argument assumes the vector spherical harmonic addition formulas quoted from the authors' earlier paper [39] are correct and valid under the convergence conditions used here; the manuscript does not prove them, and every multiplier identity and error estimate rests on them.

Editorial extensions

If this is right

  • For any scalar zonal kernel with $\hat\varphi(\ell)\asymp(1+\lambda_\ell)^{-\sigma}$ and $\sigma>1$, the $m=1$ kernel is positive definite on tangent data and its native space is norm-equivalent to $H^\sigma_{\rm div}(S^2)$, so all the scalar kernel's smoothness is available for divergence-free interpolation.
  • The interpolation matrix in local tangent frames satisfies $\lambda_{\min}(A_{K,X})\ge C q_X^{2\sigma-2}$ and $\operatorname{cond}_2(A_{K,X})\le C q_X^{-2\sigma}$, so stability degrades exactly as it would for a scalar kernel of order $\sigma$, not $\sigma-1$.
  • For targets $f\in H^\tau_{\rm div}$ with $1<\tau\le\sigma$, the interpolant satisfies $\|f-I_X f\|_{H^s}\le C\rho_X^{\sigma-\tau}h_X^{\tau-s}\|f\|_{H^\tau}$ for $0\le s\le\tau$, and targets in the native space converge at the full $\sigma$ rate in $L^\infty$.
  • Extra-smooth targets produce superconvergence: for $g\in F^{1+\vartheta}_{\rm div}$, $\|g-I_Xg\|_{N_K}\le C h_X^{\vartheta\sigma}\|g\|_{F^{1+\vartheta}_{\rm div}}$ and $\|g-I_Xg\|_{H^s}\le C h_X^{(1+\vartheta)\sigma-s}$ for $0\le s\le\sigma$; smooth test fields on $S^2$ show the predicted high rates, for example roughly $O(h^{11})$ for the $m=0$ variant with the Matern 7/2 kernel.
  • Because the construction uses the normal cross-product and tangent projection, the same kernel formula applies to any smooth oriented embedded surface; the numerical examples on a torus, a red blood cell surface, and a bumpy sphere confirm tangency and divergence-free reconstruction outside the sphere.

Reading between the lines

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

  • The $m=0$ variant, although lacking a general positive-definiteness theorem here, converges fastest numerically (about $O(h^{11})$ on smooth fields with the Matern kernel). One testable conjecture the paper leaves open is that for scalar kernels whose radial derivatives satisfy stronger positivity conditions, the $m=0$ construction achieves a native space of order $\sigma+1/2$ or $\sigma+1$; a dir
  • The inverse Laplace--Beltrami construction effectively transplants scalar Fourier multipliers onto the divergence-free subspace. This suggests a general recipe for other two-point homogeneous spaces where vector addition formulas exist; the paper does not pursue it, but the same cancellation of the $\lambda_\ell$ factor should work there.
  • The slower convergence on the bumpy sphere suggests that strong curvature variation breaks the clean $H^\sigma_{\rm div}$ native-space equivalence. A plausible extension, which the paper lists as future work, is a curvature-aware normalization of the kernel; one could test this by computing the interpolation error on a family of surfaces interpolating between the sphere and increasingly oscillator
  • The observed smallest eigenvalues follow the exponents $2\sigma-2$ predicted for $m=1$ and $m=0$ variants, so the Fourier-localization proof likely captures the true spectral behavior. A matching upper bound for $\lambda_{\min}$ remains open and would round out the stability picture.
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

3 major / 4 minor

Summary. The paper develops a family of divergence-free tangential matrix-valued kernels on embedded surfaces, with the main analysis on the unit sphere. Starting from an isotropic ambient ansatz for curl-free kernels, the authors construct surface kernels of the form Kdiv = alpha(r)((n_x·n_y)I - n_y n_x^T) - beta(r)(n_x×(x-y))(n_y×(x-y))^T, and on S^2 reduce this to Kdiv = phi^(m-1)(t)R + phi^(m)(t)Q. They compute the vector spherical harmonic multipliers (Theorem 3.1), show that the m=1 variant preserves the scalar Sobolev order while the classical m=2 variant loses one order (Corollary 3.2), and introduce an inverse Laplace-Beltrami construction whose multipliers equal those of the underlying scalar kernel (Theorem 3.4). For interpolation at scattered nodes they prove a lower bound on the smallest eigenvalue of the interpolation matrix (Theorem 5.2), pointwise and fractional Sobolev error estimates (Theorems 5.8 and 5.11), and a Hilbert-scale superconvergence result (Theorem 5.13). Numerical experiments on S^2 and on three non-spherical surfaces illustrate convergence and stability, including a K0_div variant that is explicitly outside the proved theory.

Significance. If the results hold, the main contribution is substantial: for a scalar zonal kernel with Fourier coefficients decaying like l^{-2σ}, the m=1 construction gives divergence-free Fourier multipliers also decaying like l^{-2σ}, so the native space is H^σ_div, one Sobolev order smoother than the classical m=2 construction. The multiplier-preserving inverse Laplace-Beltrami construction is elegant and potentially useful. The paper also provides a clean Fourier-based stability proof and fractional-order Sobolev error estimates, including superconvergence for targets smoother than the native space. Strengths include the explicit multiplier computations, the direct proof strategy for Theorem 3.1, and the fact that the numerical rates for K1_div and K2_div match the stated exponents, with K2_div reproducing values from the earlier literature. The main caveats are the unproved vector addition formula imported from a companion preprint and the m=0 numerical variant, which the paper itself acknowledges lacks a positive-definiteness or native-space theorem.

major comments (3)
  1. [Section 2.1, Lemma 2.1] The vector spherical harmonic addition formulas for S_l and T_l are cited from the companion preprint [39] and are not proved in this manuscript. These formulas are the keystone of the Fourier analysis: they are used in the proof of Theorem 3.1 to obtain the multiplier identity (3.5), in the m=2 comparison, in Corollary 3.2 via (3.7), and in Theorem 3.4. If a term, sign, or normalization in the formulas for Q, R, V, or W were incorrect, the central one-order gain and the multiplier-preserving property would collapse. Please include a proof, or at least a complete self-contained derivation in an appendix, and state explicitly the hypotheses under which the formulas hold for the series manipulations performed in Theorem 3.1.
  2. [Section 3.1 and Remark 3.3] The manuscript advertises lower-order variants, including m=0, but Remark 3.3 explicitly states that no general positive-definiteness or native-space theorem is established for m=0. Nevertheless, the text after (3.3) says that these cases still produce positive definite divergence-free kernels, and Section 6 presents K0_div convergence rates (Tables 1 and 2) and eigenvalue fits (Figure 4, with sigma=11/2) as if they were consequences of the theory. Since Theorem 5.13 and Corollary 5.5 require Assumption 5.1, which is unproved for K0_div, the K0 experiments are outside the paper's theorems. Either supply the missing theorem using the conditions from [38] or explicitly label the m=0 numerical results as heuristic and outside the proven framework.
  3. [Section 6, Tables 1 and 2 and Figure 4] The convergence rates reported for K0_div are presented as 'consistent with the Hilbert-scale superconvergence estimate established in Theorem 5.13', but Theorem 5.13 presupposes the multiplier condition (2.7) and positive multipliers, which are not established for K0_div. The same applies to the smallest-eigenvalue exponents sigma=11/2, 9/2, 7/2 in Figure 4; only the K1 and K2 cases are covered by the theorems in this paper. The text should distinguish clearly between proved rates and rates inferred from numerical fits, especially because the abstract emphasizes lower-order variants.
minor comments (4)
  1. [Section 6.3, RBC paragraph] The paragraph on the red blood cell surface says the interpolant is 'constructed using the proposed multiplier-preserving kernel K0_div', but K0_div was defined in Section 6 as the kernel with beta_0(r)=phi(r), whereas the multiplier-preserving construction is the one in Theorem 3.4 with K = psi'(t)Q + psi(t)R. Please correct the terminology or the kernel definition.
  2. [Section 3.1, sentence after (3.3)] The sentence 'As we show below, these cases still produce positive definite divergence-free kernels' is too strong given Remark 3.3, which disclaims a theorem for m=0. Please rephrase to state that the m=1 and m=2 cases are proved and that the m=0 case is formal and only supported numerically.
  3. [Section 5.3, Lemma 5.10] The proof invokes [16, Theorem 4.1] and asserts that it is stated for any real tau>1. Since the fractional-order Sobolev estimates in Theorem 5.11 depend on this lemma, please quote the precise statement of that theorem or provide a proof, so that the reader can verify the fractional regularity range.
  4. [Throughout] The manuscript contains several OCR-like artifacts: 'Fourier ana lysis' in the paragraph before Lemma 2.1, 'greaterorsimilar' in the stability discussion of Section 6.2, and irregular spacing in the title and abstract. Please clean these formatting issues before publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the multiplier identities are derived from scalar Legendre expansions and the mean value theorem; the self-cited Lemma 2.1 is a parameter-free addition formula, not the target result.

full rationale

The central derivation chain is self-contained in the relevant sense: (3.3) defines Kdiv from the scalar zonal generator; Theorem 3.1 expands phi^(m-1) in Legendre polynomials, uses Legendre identities to rewrite the series in terms of P'_ell, differentiates, and applies the vector addition formulas to obtain the multipliers (3.5); Corollary 3.2 obtains kappa_ell as (1+lambda_ell)^(-sigma) for m=1 by the mean value theorem applied to the explicit scalar model h(ell)=c(1+ell(ell+1))^(-sigma); and Theorem 3.4 verifies its multiplier-preserving claim directly from the spectral definition of u. None of these steps fits a parameter to the predicted quantity, nor does any output quantity enter the hypotheses by definition. The only self-cited ingredient, Lemma 2.1 from [39], is a parameter-free identity about the vector spherical harmonics defined in (2.2); it does not assume the one-order gain, the native-space equivalence, or any other central conclusion, and it is directly checkable by finite-dimensional calculation. Under the stated rules, such a parameter-free cited identity counts as real evidence rather than circularity. The m=0 case is explicitly labelled as not covered by a general theorem in Remark 3.3, with its dependence on [38] disclosed, and the paper presents it as a numerical variant rather than a proven central result. Numerical experiments are reported as corroboration, not as proof, and the theoretical rates are derived from the Fourier multipliers. The skeptical concern that Lemma 2.1 is unproved here is a correctness or completeness risk, not a circularity of the derivation chain.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The theoretical core has no fitted constants; the main free parameters are hand-chosen shape parameters in experiments. The most consequential unproved inputs are the vector addition formulas from the authors' companion paper and the imported m=0 positive-definiteness and decay assumptions.

free parameters (3)
  • Shape parameter eps for Matérn MA7/2 experiments = 9, 7, 4 for K0, K1, K2
    Hand-chosen in Section 6.1; affects observed convergence rates but not the theoretical multipliers.
  • Shape parameter eps for Wendland WE3,3 experiments = 11/10, 4/3, 5/3 for K0, K1, K2
    Hand-chosen in Section 6.1; used for the convergence tables.
  • Shape parameter eps in stability and surface examples = 4/3, 2, 7/3, 5 depending on test
    Hand-chosen in Sections 6.2 and 6.3 for Wendland and Matérn kernels.
assumptions (6)
  • standard math Vector spherical harmonic addition formulas S_ell and T_ell in Lemma 2.1 hold with the stated Q, R, V, W matrices.
    Invoked in Sections 3.2 and Appendix A; cited to companion paper [39] rather than proved here.
  • domain assumption The surface kernel (3.1), obtained by applying X_{n_x} and P_{n_y} to an ambient curl-free kernel, is tangential and surface divergence-free on arbitrary smooth oriented surfaces.
    Used in Section 3.1 and the nonspherical experiments; carried from [31, Theorem 1] with no self-contained proof.
  • domain assumption Assumption 5.1: positive multipliers with two-sided decay kappa_ell satisfies c1(1+lambda_ell)^{-sigma} <= kappa_ell <= c2(1+lambda_ell)^{-sigma}, sigma > 1.
    This is the working hypothesis for the stability and error theorems in Sections 5.1 through 5.4.
  • ad hoc to paper The m=0 K0 kernels satisfy the positive-definiteness and multiplier-decay assumptions used in the numerical convergence discussion.
    Remark 3.3 explicitly withholds a general positive-definiteness theorem for m=0; Section 6 still treats K0 as covered, importing conditions from the authors' [38].
  • standard math Standard localization estimates for filtered spherical harmonic kernels and scalar Bernstein inequalities hold as quoted.
    Used in proofs of Theorem 5.2 and Lemma 5.7, quoted from [4,5,26].
  • standard math Fractional sampling inequalities and Gagliardo-Nirenberg interpolation on S2 hold as quoted.
    Used in Lemma 5.9, quoted from [2,3].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Divergence-free interpolation of tangential vector fields via matrix-valued kernels." pith.science (2026). https://pith.science/paper/RLQANEBK

@misc{pith2026260805547,
  author       = {Pith},
  title        = {Pith review of: Divergence-free interpolation of tangential vector fields via matrix-valued kernels},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RLQANEBK}},
  note         = {Machine review of arXiv:2608.05547}
}
read the original abstract

We develop and analyze a family of divergence-free kernel interpolation methods for tangential vector fields on the unit sphere. Starting from scalar radial kernels in Euclidean space, we construct tangent-valued, surface divergence-free matrix kernels without repeatedly applying surface differential operators. The construction separates the geometric enforcement of the divergence-free constraint from the choice of scalar generator and admits lower-order variants with reduced regularity requirements compared with classical potential-based methods. Using vector spherical harmonics, we derive explicit kernel representations and characterize their Fourier multipliers. We also introduce an inverse Laplace-Beltrami construction that preserves the multipliers of the underlying scalar zonal kernel. For interpolation at scattered nodes, we establish a lower bound for the smallest eigenvalue of the interpolation matrix and derive pointwise and Sobolev error estimates, including superconvergence for targets smoother than the native space. Numerical experiments corroborate the theoretical convergence and stability results, while additional examples on nonspherical surfaces illustrate the applicability of the kernel formula beyond the sphere.

Figures

Figures reproduced from arXiv: 2608.05547 by the authors.

Figure 1
Figure 1. Relative ℓ2 errors for Field 2 obtained via matrix-valued kernel interpolation with MA7/2 and WE3,3 kernels on ME node sets. kernels K0 div, K1 div, and K2 div. Among them, K0 div yields the smallest errors across the sphere, whereas K2 div produces the largest. 6.2. Spectral stability of the interpolation matrices. We assess the spectral stability of the interpolation matrices by examining their smallest eigenvalue… view at source ↗
Figure 2
Figure 2. Comparison of the target and reconstructed fields. Top row: Field 1. (a) Stream function s1; (b) divergence-free vector field L∗s1; and (c) reconstructed field using K0 div with N = 1849 ME nodes. Bottom row: Field 2. (d) Stream function s2; (e) divergence-free vector field L∗s2; and (f) reconstructed field using K0 div with N = 1849 ME nodes. (a) K0 div (b)K1 div (c) K2 div [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Pointwise errors for approximating Field 1 using the three matrix￾valued kernels. All experiments use N = 1849 ME interpolation nodes and M = 20164 evaluation nodes. The top row of [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Minimum eigenvalues of the divergence-free interpolation matrices. Top: K0 div, K1 div, K2 div with MA7/2 on ME (left) and Fibonacci (right) nodes. Bottom: K0 div with Wendland kernels ϕ3,3, ϕ5,3, ϕ7,3 on ME (left) and Fi￾bonacci (right) nodes. For the bumpy sphere, we…
Figure 5
Figure 5. Figure 5: Minimum eigenvalues of the multiplier-preserving divergence-free interpolation matrices Kdiv(x, y) = ψ ′ (t)Q(x, y) + ψ(t)R(x, y) constructed from the Wendland kernels ϕ3,3, ϕ5,3, and ϕ7,3 with the common shape pa￾rameter ε = 2. Results are shown for ME nodes (left) an…
Figure 6
Figure 6. Figure 6: Stream functions and reconstructed divergence-free vector fields on three embedded surfaces: torus, red blood cell and bumpy sphere. All reconstructions are obtained via K0 div . [4] G. Brown and F. Dai. Approximation of smooth functions on compact two-point homogeneou…
Figure 7
Figure 7. Figure 7: Relative ℓ2 errors of the divergence-free approximation on three different surfaces using the restricted MA7/2 kernel. [10] P. Farrell, K. Gillow, and H. Wendland. Multilevel interpolation of divergence-free vector fields. IMA J. Numer. Anal., 37(1):332–353, 2017. [11]…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

50 extracted references · 43 canonical work pages

  1. [39]

    Z.J. Sun, B. Huang, and X.P. Sun. Vector field multiplier operators and matrix-valued kernel quasi- interpolation. arXiv:2605.05610, 2026

    work page Pith review arXiv 2026

  2. [38]

    Error estimates for vector field interpolation based on generalized matrix-valued kernels

    Z.J. Sun, L.S. Dong, and L. Ling. Error estimates for vec tor field interpolation based on generalized matrix- valued kernels. arXiv:2608.04313, 2026

    work page Pith review arXiv 2026

  3. [1]

    Adams and J.J.F

    R.A. Adams and J.J.F. Fournier. Sobolev Spaces, 2nd edition. Vol. 140. of Pure and Applied Mathematics, Elsevier/Academic Press, Amsterdam, 2003

    work page 2003

  4. [2]

    Arcang´ eli, M.C

    R. Arcang´ eli, M.C. L´ opez de Silanes, and J.J. Torrens. Extension of sampling inequalities to Sobolev semi- norms of fractional order and derivative data. Numer. Math. , 121(3):587–608, 2012

    work page 2012

  5. [3]

    Brezis and P

    H. Brezis and P. Mironescu. Gagliardo-Nirenberg inequa lities and non-inequalities: the full story. Ann. Inst. Henri Poincar´ e, Anal. Non lin´ eaire, 35(5):1355–1376, 2018. DIVERGENCE-FREE INTERPOLATION 25 (a) Stream function s3 (b) Reconstructed vector field (c) Stream function s4 (d) Reconstructed vector field (e) Stream function s5 (f) Reconstructed vec...

    work page 2018

  6. [4]

    Brown and F

    G. Brown and F. Dai. Approximation of smooth functions on compact two-point homogeneous spaces. J. Func. Anal., 220(2):401–423, 2005

    work page 2005

  7. [5]

    Dai and Y

    F. Dai and Y. Xu. Approximation theory and harmonic analysis on spheres and b alls, volume 23. Springer, 2013

    work page 2013

  8. [6]

    Dodu and C

    F. Dodu and C. Rabut. Irrotational or divergence-free in terpolation. Numer. Math. , 98(3):477–498, 2004

    work page 2004

Show all 50 references
  1. [7]

    Drake, E.J

    K.P. Drake, E.J. Fuselier, and G.B. W right. A partition o f unity method for divergence-free or curl-free radial basis function approximation. SIAM J. Sci. Comput. , 43(3):A1950–A1974, 2021

    2021

  2. [8]

    Drake, E.J

    K.P. Drake, E.J. Fuselier, and G.B. W right. Implicit sur face reconstruction with a curl-free radial basis function partition of unity method. SIAM J. Sci. Comput. , 44(5):A3018–A3040, 2022

    2022

  3. [9]

    F. Dubois. Discrete vector potential representation of a divergence-free vector field in three-dimensional domains: numerical analysis of a model problem. SIAM J. Numer. Anal. , 27(5):1103–1141, 1990. DIVERGENCE-FREE INTERPOLATION 26 10−2 10−1 10−9 10−7 10−5 10−3 10−1 h11 X h7 ...

    1990

  4. [10]

    Farrell, K

    P. Farrell, K. Gillow, and H. W endland. Multilevel inte rpolation of divergence-free vector fields. IMA J. Numer. Anal. , 37(1):332–353, 2017

    2017

  5. [11]

    Freeden and M

    W. Freeden and M. Schreiner. Spherical functions of mathematical geosciences . Springer, 2008

    2008

  6. [12]

    Freedman and Z.-X

    M.H. Freedman and Z.-X. He. Divergence-free fields: ene rgy and asymptotic crossing number. Ann. Math. , 134(1):189–229, 1991

    1991

  7. [13]

    Fuselier and G.B

    E. Fuselier and G.B. W right. Scattered data interpolat ion on embedded submanifolds with restricted positive definite kernels: Sobolev error estimates. SIAM J. Numer. Anal. , 50(3):1753–1776, 2012

    2012

  8. [14]

    Fuselier

    E.J. Fuselier. Improved stability estimates and a char acterization of the native space for matrix-valued RBFs. Adv. Comput. Math. , 29(3):269–290, 2008

    2008

  9. [15]

    Fuselier

    E.J. Fuselier. Sobolev-type approximation rates for d ivergence-free and curl-free RBF interpolants. Math. Comput., 77(263):1407–1423, 2008

    2008

  10. [16]

    Fuselier, F.J

    E.J. Fuselier, F.J. Narcowich, J.D. W ard, and G.B. W rig ht. Error and stability estimates for surface- divergence free RBF interpolants on the sphere. Math. Comput. , 78(268):2157–2186, 2009

    2009

  11. [17]

    Fuselier and G.B

    E.J. Fuselier and G.B. W right. Stability and error esti mates for vector field interpolation and decomposition on the sphere with RBFs. SIAM J. Numer. Anal. , 47(5):3213–3239, 2009

    2009

  12. [18]

    Guzm´ an and M

    J. Guzm´ an and M. Neilan. Conforming and divergence-fr ee Stokes elements in three dimensions. IMA J. Numer. Anal. , 34(4):1489–1508, 2014

    2014

  13. [19]

    Guzm´ an and M

    J. Guzm´ an and M. Neilan. Conforming and divergence-fr ee Stokes elements on general triangular meshes. Math. Comput. , 83(285):15–36, 2014

    2014

  14. [20]

    H. Hermes. Nilpotent and high-order approximations of vector field systems. SIAM Rev. , 33(2):238–264, 1991

    1991

  15. [21]

    Karvonen, G

    T. Karvonen, G. Santin, and T. W enzel. General supercon vergence for kernel-based approximation. arXiv:2505.11435, 2025

    2025 arXiv

  16. [22]

    Lederer, C

    P.L. Lederer, C. Lehrenfeld, and J. Sch¨ oberl. Diverge nce-free tangential finite element methods for incom- pressible flows on surfaces. Int. J. Numer. Methods Eng. , 121(11):2503–2533, 2020

    2020

  17. [23]

    Lowitzsch

    S. Lowitzsch. Error estimates for matrix-valued radia l basis function interpolation. J. Approx. Theory , 137(2):238–249, 2005

    2005

  18. [24]

    Lowitzsch

    S. Lowitzsch. Matrix-valued radial basis functions: s tability estimates and applications. Adv. Comput. Math., 23(3):299–315, 2005

    2005

  19. [25]

    H.N. Mhaskar. On the representation of smooth function s on the sphere using finitely many bits. Appl. Comput. Harmon. Anal. , 18(3):215–233, 2005

    2005

  20. [26]

    Mhaskar, F.J

    H.N. Mhaskar, F.J. Narcowich, J. Prestin, and J.D. W ard . Lp Bernstein estimates and approximation by spherical basis functions. Math. Comput. , 79(271):1647–1679, 2010

    2010

  21. [27]

    Morton and M

    T.M. Morton and M. Neamtu. Error bounds for solving pseu dodifferential equations on spheres by colloca- tion with zonal kernels. J. Approx. Theory , 114(2):242–268, 2002

    2002

  22. [28]

    M¨ uller

    C. M¨ uller. Spherical Harmonics . Lecture Notes in Mathematics, Vol. 17, Springer-Verlag, B erlin, 1966

    1966

  23. [29]

    Narcowich, X.P

    F.J. Narcowich, X.P. Sun, J.D. W ard, and H. W endland. Di rect and inverse Sobolev error estimates for scattered data interpolation via spherical basis function s. Found. Comput. Math. , 7(3):369–390, 2007

    2007

  24. [30]

    Narcowich and J.D

    F.J. Narcowich and J.D. W ard. Generalized Hermite inte rpolation via matrix-valued conditionally positive definite functions. Math. Comput. , 63(208):661–687, 1994

    1994

  25. [31]

    Narcowich, J.D

    F.J. Narcowich, J.D. W ard, and G.B. W right. Divergence -free RBFs on surfaces. J. Fourier Anal. Appl. , 13(6):643–663, 2007. DIVERGENCE-FREE INTERPOLATION 27

    2007

  26. [32]

    Neilan and B

    M. Neilan and B. Otus. Divergence-free Scott-Vogelius elements on curved domains. SIAM J. Numer. Anal., 59(2):1090–1116, 2021

    2021

  27. [33]

    Schaback

    R. Schaback. Improved error bounds for scattered data i nterpolation by radial basis functions. Math. Com- put., 68(255):201–216, 1999

    1999

  28. [34]

    Schaback and Z.M

    R. Schaback and Z.M. W u. Operators on radial functions. J. Comput. Appl. Math. , 73(1-2):257–270, 1996

    1996

  29. [35]

    Schwarzacher, B.W

    S. Schwarzacher, B.W. She, and K. T ˚ uma. Stability and e rror estimates of a linear numerical scheme approximating nonlinear fluid-structure interactions. Numer. Math. , 157(3):1023–1077, 2025

    2025

  30. [36]

    K. Soga. Mathematical analysis of a finite difference met hod for inhomogeneous incompressible Navier– Stokes equations. Numer. Math. , 156(5):1809–1853, 2024

    2024

  31. [37]

    H.A. Stone. Interfaces: in fluid mechanics and across di sciplines. J. Fluid Mech. , 645:1–25, 2010

    2010

  32. [40]

    Sun and L

    Z.J. Sun and L. Ling. A kernel-based meshless conservat ive Galerkin method for solving Hamiltonian wave equations. SIAM J. Sci. Comput. , 44(4):A2789–2807, 2022

    2022

  33. [41]

    Swarztrauber

    P.N. Swarztrauber. The approximation of vector functi ons and their derivatives on the sphere. SIAM J. Numer. Anal. , 18(2):191–210, 1981

    1981

  34. [42]

    W endland

    H. W endland. Piecewise polynomial, positive definite a nd compactly supported radial functions of minimal degree. Adv. Comput. Math. , 4(1):389–396, 1995

    1995

  35. [43]

    W endland

    H. W endland. Scattered data approximation , volume 17. Cambridge University Press, 2004

    2004

  36. [44]

    W endland

    H. W endland. Divergence-free kernel methods for appro ximating the Stokes problem. SIAM J. Numer. Anal., 47(4):3158–3179, 2009

    2009

  37. [45]

    W u and C.-W

    K.L. W u and C.-W. Shu. Provably physical-constraint-p reserving discontinuous Galerkin methods for mul- tidimensional relativistic MHD equations. Numer. Math. , 148(3):699–741, 2021

    2021

  38. [46]

    Zhang, K

    E. Zhang, K. Mischaikow, and G. Turk. Vector field design on surfaces. ACM Trans. Graph. (ToG) , 25(4):1294–1326, 2006

    2006

  39. [47]

    J.K. Zhao, B. Zhang, S.P. Mao, and S.C. Chen. The diverge nce-free nonconforming virtual element for the Stokes problem. SIAM J. Numer. Anal. , 57(6):2730–2759, 2019. Appendix A. Proof of Lemma 5.3 To prove Lemma 5.3, we first recall a result about uniform filtered Ja cobi local...

    2019

  40. [48]

    Let Γ T ⊂ R3 be the torus parameterized by xT(θ,λ ) = ( (R +r cosλ) cosθ, (R +r cosλ) sinθ, rsinλ )⊤ , θ,λ ∈ [0, 2π), where R and r denote the major and minor radii, respectively

    T orus. Let Γ T ⊂ R3 be the torus parameterized by xT(θ,λ ) = ( (R +r cosλ) cosθ, (R +r cosλ) sinθ, rsinλ )⊤ , θ,λ ∈ [0, 2π), where R and r denote the major and minor radii, respectively. We generate a tang ential divergence-free vector field from the stream function s3(θ,λ ) =...

  41. [49]

    Red blood cell. The RBC surface Γ R ⊂ R3 is parameterized by ΓR = { (x,y,z ) ∈ R3 |x =r0 cosλ cosθ, y =r0 sinλ cosθ, z = 1 2 sinθ ( c0 +c2 cos2θ +c4 cos4θ )} , where −π/2 ≤ θ ≤ π/2, −π ≤ λ < π, r0 = 3.91/3.39, c0 = 0.81/3.39, c2 = 7.83/3.39, and c4 = −4.39/3.39. This biconcave...

  42. [50]

    Bumpy sphere. The bumpy sphere Γ B ⊂ R3 is defined as a radial perturbation of the unit sphere: xB(θ,λ ) = ρ(θ,λ )   sinλ cosθ sinλ sinθ cosλ  , θ ∈ [0, 2π), λ ∈ [0,π ], where the radial function is given by ρ(θ,λ ) = Rbase +a [sin(ωx) + sin(ωy) + sin(ωz)], with (x,y,z ) = ...

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.