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REVIEW 3 major objections 6 minor 29 references

Gibbs Phenomena for $L^q$-Best Approximation in Finite Element Spaces -- Some Examples

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Pith's one-line read This paper identifies which finite element meshes retain the Gibbs overshoot in $L^q$-best approximation, and shows that as $q\to1$ the overshoot vanishes on some meshes while persisting on others; the deciding factor is the relative size…

desk verdict A solid, honest set of worked examples showing when L1-best approximation can avoid Gibbs overshoot in FE spaces; the q→1 limit claims, however, outrun the proofs in several of the advertised positive cases. read the letter →

arxiv 1909.00658 v1 pith:WT64R5R6 submitted 2019-09-02 math.NA cs.NA

Paul Houston , Sarah Roggendorf , Kristoffer G. van der Zee This is my paper · ORCID
classification math.NAcs.NA MSC 41A5065N30
keywords Lq-bestapproximationGibbsphenomenonfiniteelementspacespiecewiselinearL1minimizationmeshgradingnon-uniquenessofbest
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 asks whether the $L^q$-best approximation of a discontinuous function by continuous piecewise-linear finite element functions can be made free of over- and undershoots by letting $q$ tend to 1. Using simple one- and two-dimensional examples that can be solved in closed form, it shows that the answer is yes on meshes satisfying explicit structural conditions, such as elements near the discontinuity not being too small relative to their neighbours toward the discontinuity. On other meshes, including some uniform meshes in two dimensions, a nonzero overshoot survives even at $q=1$, although its size decreases as $q\to1$. The motivation is the prospect of designing finite element methods in $L^1$-type spaces, where spurious oscillations near layers and shocks would be absent by construction. If the claims hold, choosing $L^1$ together with a suitably designed mesh becomes a tool for suppressing Gibbs phenomena rather than a source of instability.

What carries the argument

The machinery is the subdifferential characterization of best approximation: $u_h$ minimizes $\|u-w\|_{L^q}$ iff $\int_\Omega \operatorname{sgn}(u-u_h)|u-u_h|^{q-1}v_h\,dx=0$ for every test function $v_h$ in the finite element space. For $q=1$ the sign function on the set where $u=u_h$ is free in $[-1,1]$, which produces families of $L^1$-minimizers and lets the paper construct a sign $\psi$ that annihilates every hat function. Each theorem is proved by splitting the residual integral over the parts of each element where $u-u_h$ is positive and negative, reducing the problem to sign-balance equations whose solutions are the stated nodal values.

What would settle it

Compute the $L^q$-minimizers for the symmetric four-element jump mesh with $h=0.75$ for $q=1.1,1.01,1.001,\ldots$; if $u_h(h)$ does not approach $\sqrt{2h}\approx1.2247$ with $u_h(0)=0$, the convergence claim in Theorem 1.4(4) is false. The same check at $h=0.4$ should see $u_h(h)\to1$.

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Extended reading notes

Core claim

For the boundary discontinuity $u\equiv1$ on $(0,1)$ with $u_h(0)=1$, $u_h(1)=0$, the paper derives the full $L^q$-best approximation on a two-element mesh: the nodal value $\alpha$ satisfies an explicit algebraic equation for $1<q<\infty$, and at $q=1$ it is $\alpha=1$ when the element touching the outflow boundary has length $h\le1/2$, otherwise $\alpha=\sqrt{2h}$, so the overshoot persists exactly when that element is longer than the rest. On $N$-element meshes, the paper proves sufficient conditions, in particular a graded-mesh inequality and the simpler condition that the last element is not longer than any earlier element, under which an $L^1$-best approximation with no over- or undershoot exists. For the interior jump $\operatorname{sgn}(x)$ on a symmetric four-element mesh, there is a one-parameter family of $L^1$-best approximations, and the paper claims that the $L^q$-minimizers converge as $q\to1$ to the odd member of this family, whose overshoot is $1$ for $h\le1/2$ and $\sqrt{2h}$ for $h>1/2$. In two dimensions, the paper analyses four meshes: the uniform Mesh 1 has a unique $L^1$-best approximation with overshoot $\alpha\approx1.3200$, Mesh 2 has overshoot $\alpha\approx1.2723$, and Meshes 3 and 4 admit $L^1$-best approximations with all interior node values equal to 1, so the presence or absence of Gibbs phenomena at $q=1$ is settled by the mesh geometry.

Load-bearing premise

The argument assumes that the unique best approximation for each $q>1$ moves continuously as $q$ approaches 1 and lands on an $L^1$ best approximation, with the odd symmetry of every $L^q$-minimizer passed to the limit; no proof of this convergence is given.

Editorial extensions

If this is right

  • If the mesh satisfies the stated grading conditions, an $L^1$-best approximation of a boundary discontinuity exists whose nodal values never exceed the data, so the approximation is oscillation-free at $q=1$.
  • On the symmetric four-element jump mesh, the limit $q\to1$ selects the odd $L^1$-minimizer; the overshoot disappears when the elements adjacent to the jump are no longer than the outer elements, and every $L^1$-minimizer overshoots otherwise.
  • In two dimensions, Mesh 1 and Mesh 2 show that even uniform or structured meshes can retain a fixed overshoot at $q=1$, and uniform refinement of Mesh 1 keeps the overshoot constant, so mesh design rather than refinement is the lever for removing Gibbs phenomena.
  • On Meshes 3 and 4 there exist oscillation-free $L^1$-best approximations, demonstrating that the two-dimensional analogue of the one-dimensional grading condition is realisable.
  • On all meshes considered, the magnitude of the over- and undershoots decreases as $q\to1$, so even when the overshoot persists it is smaller in $L^1$ than in $L^2$.

Reading between the lines

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

  • Editorial inference: the same sign-balance argument should carry over to higher-order elements and to $L^p$ with $p$ slightly above 1, where the overshoot size would be controlled by the $q$-weighted residual near the discontinuity; a direct computation on quadratic elements would test this.
  • Editorial inference: the unproved convergence of $L^q$-minimizers to $L^1$-minimizers in Theorem 1.4(4) could be replaced by an explicit selection principle, such as the minimizer with minimal energy within the $L^1$-minimizer family, and verified numerically; if the limit instead depends on the path $q\to1$, the symmetric-jump conclusion needs qualification.
  • Editorial inference: the two-dimensional area-balance criterion used in Section 6.3.2, comparing the area of triangles touching a node and the outflow boundary with the area of the remaining triangles around that node, is presented as a design heuristic rather than a theorem; proving it for general triangulations would give a practical mesh-generation rule for oscillation-free $L^1$ approximation.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper studies the Lq-best approximation (1≤q<∞) of discontinuous functions—u≡1 with inflow/outflow boundary conditions, and u(x)=sgn(x)—by continuous piecewise-linear finite element functions on selected one- and two-dimensional meshes. For 1<q<∞ the minimizer is unique and characterized by the subdifferential condition (2.2); for q=1 the paper uses the multivalued subdifferential to characterize the generally nonunique L1 minimizers. The main theorems give explicit nodal values for the free parameters on the two-element mesh (Theorem 1.1(1)), the symmetric four-element mesh (Theorem 1.4), and four structured two-dimensional meshes (Theorem 1.5). The stated goal is to show that over- and undershoots (Gibbs phenomena) vanish in the limit q→1 on meshes satisfying certain structural conditions, while persisting on others, and that the behavior is mesh-dependent.

Significance. The paper contains detailed, apparently correct computations of the relevant minimizers for the model problems; the two-element 1D case and the Mesh-1 2D case are fully worked out, and the polynomial equations are explicit enough to be re-derived by the reader. The findings provide useful, falsifiable predictions: graded 1D meshes with h_N≤min h_i and certain two-dimensional 'balanced area' meshes admit L1-best approximations without over/undershoots, while uniform 2D Mesh 1 and some nonuniform 1D meshes do not. The L1 characterization via the Singer subdifferential is applied carefully, and the paper is not circular: it does not rely on the authors' prior results beyond motivation. However, the advertised q→1 limit statements are not supported by the proofs as written; the existence of a favorable L1 minimizer is weaker than convergence of the Lq minimizers to a favorable limit.

major comments (3)
  1. [Section 4.3 (proof of Theorem 1.4(4))] The convergence claim in part 4 is asserted without proof. The text says that since every Lq minimizer is odd, 'the limit as q→1 must be an odd function as well' and therefore equals the anti-symmetric L1 minimizer with uh(0)=0. This presupposes (i) that the family {u_q} has a limit, (ii) that any cluster point is an L1 minimizer, and (iii) that the L1 minimizer set contains a unique odd element. For h≤0.5, Theorem 1.4(1) gives a continuum of L1 minimizers parameterized by β∈[−1,1], so (iii) alone is insufficient; the argument needs to prove that every cluster point is an odd L1 minimizer, which forces β=0, and that the family is relatively compact (e.g., uniformly bounded). None of these steps appear. Since part 4 is the paper's only explicit statement of convergence to a Gibbs-free limit for the jump example, this is a load-bearing gap. The claim is likely repairable by the compactness argument indicated, but as written it is unproven.
  2. [Theorems 1.1(2,3), 1.5(4,5); abstract and Section 7] The abstract and conclusions state that on certain meshes the Gibbs phenomenon 'can be eliminated in the limit as q tends to 1'. However, Theorems 1.1(2,3) and 1.5(4,5) only construct an L1-best approximation with no over- or undershoots; they do not show that the unique Lq-minimizers (q>1) converge to such a minimizer as q→1. Because L1 minimizers are nonunique in some of these examples (Theorem 1.4(1)), the existence of one favorable L1 minimizer does not imply that the Lq limit is favorable. For Theorem 1.5 the paper relies on numerical evidence (Section 6.3.1, Figs. 14–15); for Theorem 1.1(2,3) no convergence argument or numerical demonstration of the q→1 limit is given (Fig. 13 shows q=2 and q=1, not the approach to q=1). Even in the two-element case of Theorem 1.1(1), the assertion in the text that α→1 as q→1 for h≤0.5 is observed from Fig. 2 but not proved; the algebraic equation defining α(q) changes form at q=1, so a separate argument is needed. The advertised claims should either be proved (e.g., by showing relative compactness of {u_q} and that all cluster points are no-overshoot L1 minimizers) or weakened to statements about existence of L1 minimizers with the desired properties.
  3. [Section 5.3 (proof of Theorem 1.5(3))] The proof of part 3 is not valid as written. It begins 'if there are no over- or undershoots, i.e., uh(0.5,1)=uh(0.5,0.5)=uh(0.5,0)=1'. The 'i.e.' is incorrect: absence of over- or undershoots only requires 0≤uh≤1, so the three free nodal values may lie strictly between 0 and 1. The subsequent positivity argument rules out only the particular candidate with all three values equal to 1; it does not exclude, for example, a function with nodal values 0.8, 0.9, 0.95 that still satisfies the boundary conditions. To prove that every Lq-best approximation for q>1 has an over- or undershoot, one must show that the optimality system (2.2) has no solution with max_i{α,β,γ}≤1; no such argument is given.
minor comments (6)
  1. [Section 2 (Corollary 2.4)] The statement says 'Let U := Lq(Ω)', but the corollary characterizes L1-best approximations; it should read L1(Ω).
  2. [Section 3.1 (first paragraph)] The boundary conditions are stated as 'uh(0)=0 and uh(1)=1', but the problem (1.3) and the constructed function uh=ϕ0+αϕ1 satisfy uh(0)=1, uh(1)=0; the text reverses them.
  3. [Theorems 1.1(2) and (3)] The mesh notation '0 = x0 < x2 <···' should read '0 = x0 < x1 < x2 <···'.
  4. [Section 5.3 (proof of part 5)] The phrase 'for the second and third mesh, α=β=γ=1 is an L1-best approximation' appears to refer to Meshes 3 and 4 in the statement of Theorem 1.5; the numbering is inconsistent.
  5. [Section 3.1 (computation for α=1)] In the displayed integral, the factor 'x+1' should be 'x/(1−h)' (the hat function φ1 on (0,1−h)); the resulting value −h/2 is nevertheless correct.
  6. [Section 6.3.2] The statement that the overshoot at nodes 2,3,5,6 'does not disappear' as q→1 is inferred from q=1.2 numerics and from the Mesh-2 analogy; it would be more precise to say the numerical evidence is consistent with that conclusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the results are derived from first-principles optimality conditions with no fitted parameters or load-bearing self-citation.

full rationale

The paper contains no circular derivation. The central results are obtained by applying the Singer subdifferential characterization of best approximation (Corollaries 2.3 and 2.4) and directly solving the resulting integral equations for each explicit mesh. No parameter is fitted to data and then renamed as a prediction; the Lq and L1 approximation coefficients are characterized by exact equations whose solutions are then compared as q tends to 1. The self-citations to Muga–van der Zee and Houston et al. appear only as background motivation or as a downstream application, not as premises in the proofs, so they are not load-bearing. The most fragile point is the assertion in Section 4.3 that the odd Lq minimizers converge as q tends to 1 to the odd L1 minimizer; the limit is justified only by an oddness argument and not by an explicit compactness or continuity proof. Similarly, the two-dimensional claims that overshoot vanishes as q tends to 1 on Meshes 3 and 4 are supported partly by numerical evidence in Section 6.3.1. These are correctness or proof-gap concerns, not circularity: the paper does not assume the conclusion it is trying to establish, and no equation is equivalent to its own input by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data and no new physical or mathematical entities. The mesh parameter h and nodal values alpha, beta, gamma are problem data or unknowns determined by derived equations, not fitted constants. The main implicit input is standard convex-analysis background plus one unproved convergence assumption (q tending to 1) flagged in the axioms.

assumptions (4)
  • standard math Subdifferential characterization of best approximation (Theorem 2.1, Singer 1970): uh minimizes ||u-wh||_U iff there exists r' in the subdifferential of the norm at u-uh that annihilates Uh.
    Quoted from Singer and used throughout Sections 3 to 5 to derive the Lq and L1 best-approximation conditions.
  • standard math For 1 < q < infinity, the Lq norm is Gateaux differentiable on Lq(Omega) with the derivative given by ||w||^{1-q} sgn(w)|w|^{q-1}; for q=1, the subdifferential consists of all psi in L-infinity with |psi|<=1 a.e. and psi=sgn(w) on the set where w is nonzero.
    Standard convex analysis, cited to Cioranescu (1990); underpins Corollaries 2.3 and 2.4.
  • standard math Existence and uniqueness of minimizers for strictly convex Lq norms (1 < q < infinity) on finite-dimensional subspaces.
    Used to justify the odd-symmetry assumption in Section 4.3 and to assert unique Lq best approximations.
  • ad hoc to paper The family of unique Lq minimizers converges as q tends to 1 to an L1 minimizer, and the limit is identified by symmetry.
    Invoked in the proof of Theorem 1.4 part 4 without proof. The argument assumes existence of the limit and continuity of minimizers in q, which is plausible but not established in the text.

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Pith. "Pith review of Gibbs Phenomena for $L^q$-Best Approximation in Finite Element Spaces -- Some Examples." pith.science (2026). https://pith.science/paper/WT64R5R6

@misc{pith2026190900658,
  author       = {Pith},
  title        = {Pith review of: Gibbs Phenomena for $L^q$-Best Approximation in Finite Element Spaces -- Some Examples},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WT64R5R6}},
  note         = {Machine review of arXiv:1909.00658}
}
abstract

Recent developments in the context of minimum residual finite element methods are paving the way for designing finite element methods in non-standard function spaces. This, in particular, permits the selection of a solution space in which the best approximation of the solution has desirable properties. One of the biggest challenges in designing finite element methods are non-physical oscillations near thin layers and jump discontinuities. In this article we investigate Gibbs phenomena in the context of $L^q$-best approximation of discontinuities in finite element spaces with $1\leq q<\infty$. Using carefully selected examples, we show that on certain meshes the Gibbs phenomenon can be eliminated in the limit as $q$ tends to $1$. The aim here is to show the potential of $L^1$ as a solution space in connection with suitably designed meshes.

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