{"id":"9c9b8d2d-8b71-4848-afb6-356c325c0d3b","arxiv_id":"1909.00658","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"On certain 1D and 2D meshes, the Lq-best approximation of a discontinuity in piecewise-linear finite element spaces has over/undershoots that vanish as q tends to 1, while on other meshes they persist even at q=1.","lead":"This paper works out exact formulas for the best Lq approximation of simple discontinuous functions, like u=1 with a boundary jump, by piecewise-linear finite element functions. It shows that as q approaches 1 the overshoot near discontinuities can vanish on some carefully chosen meshes but stays present on others, which matters for designing numerical methods without spurious oscillations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The advertised q→1 vanishing of overshoot is not proven for most positive cases: the theorems construct an oscillation-free L1 minimizer but do not prove the Lq minimizers converge to it.","rationale":"The reader identified the unproved convergence in Theorem 1.4(4), and that is indeed a genuine gap. But the gap is broader: the paper's advertised conclusion, that Gibbs phenomena vanish as q→1, is not established by the theorems for the 1D boundary-discontinuity problem or for the 2D Meshes 3 and 4. Those theorems only show that an L1-best approximation without oscillation exists, not that the unique Lq minimizers converge to such an approximation. Because L1 minimizers can be nonunique, this is a real logical gap, not a cosmetic one. The paper deserves credit for careful explicit computations and for proving the q=1 existence results, so the correct verdict is CONDITIONAL: the claims about q→1 behavior should either be proven or explicitly weakened to statements about existence of oscillation-free L1 minimizers.","tokens_in":30744,"tokens_out":11628,"duration_ms":240046,"concrete_test":"For the three-element mesh h1=0.1, h2=0.5, h3=0.4 from Section 6.2 (which satisfies the sufficient condition (1.8)), compute the full set of L1 minimizers by solving the small linear program in the two unknowns (uh(x1), uh(x2)). Then evaluate the implicit equation of Theorem 1.1(1b)/(3.2) at q=1+10^-k for k=3,4,5 to see the actual limit of the Lq minimizers. If the L1 minimizer set contains a point with max(uh)>1 and the Lq limit tends to that point rather than to (1,1), the q→1 elimination claim for Theorem 1.1(3) fails. Repeat the analogous uniqueness/limit check for the 2D Meshes 3 and 4 of Theorem 1.5(5).","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The abstract claims that on certain meshes Gibbs phenomena are eliminated in the limit q→1. For Theorem 1.4(4) this limit is asserted with no proof: the text says the limit 'must be an odd function' and therefore the odd L1 minimizer, but compactness and continuity of minimizers are never shown. In finite dimensions, uniform convergence of the functionals only ensures that cluster points of minimizers are L1 minimizers; if the L1 minimizer set is not a singleton, the actual limit depends on which member is selected. Theorem 1.4 can be repaired by oddness of all cluster points, but the repair does not extend to the boundary-discontinuity cases. Theorems 1.1(2,3) and 1.5(4,5) only prove existence of an L1-best approximation without over/undershoots; they do not prove that the unique Lq minimizers converge to such a minimizer. For Meshes 3 and 4 the paper explicitly relies on numerical evidence for the q→1 approach (Section 6.3.1). Since L1 best-approximation is nonunique in these examples, existence of one nice L1 minimizer is a strictly weaker statement than vanishing of the Lq overshoot as q→1. Hence the central advertised claim is underproved exactly where it is needed.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":31010,"tokens_out":15210,"duration_ms":125481,"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":[{"comment":"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.","section":"Section 4.3 (proof of Theorem 1.4(4))"},{"comment":"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.","section":"Theorems 1.1(2,3), 1.5(4,5); abstract and Section 7"},{"comment":"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.","section":"Section 5.3 (proof of Theorem 1.5(3))"}],"minor_comments":[{"comment":"The statement says 'Let U := Lq(Ω)', but the corollary characterizes L1-best approximations; it should read L1(Ω).","section":"Section 2 (Corollary 2.4)"},{"comment":"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.","section":"Section 3.1 (first paragraph)"},{"comment":"The mesh notation '0 = x0 < x2 <···' should read '0 = x0 < x1 < x2 <···'.","section":"Theorems 1.1(2) and (3)"},{"comment":"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.","section":"Section 5.3 (proof of part 5)"},{"comment":"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.","section":"Section 3.1 (computation for α=1)"},{"comment":"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.","section":"Section 6.3.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid contribution to the understanding of L1-best approximation in finite element spaces, and the explicit computations are valuable. The central limit claim, however, is overreaching: the missing convergence proofs are likely repairable by standard compactness arguments, but the authors should either add them or soften the claims to give existence of favorable L1 minimizers. The numerical sections are suggestive but not a substitute for the missing analysis."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my take on Houston–Roggendorf–van der Zee. The paper is a careful, honest exploration of when Lq-best approximation in continuous piecewise-linear spaces loses Gibbs overshoot as q→1. What is genuinely new is the systematic treatment of fixed meshes with boundary conditions: two-element and N-element 1D meshes, a symmetric four-element jump example, and several 2D meshes. The subdifferential characterizations are applied cleanly, the integral computations are explicit, and I verified the key two-element and Mesh-1 equations. The negative examples (uniform 2D Mesh 1, some non-uniform 1D meshes) are as important as the positive ones, and they keep the paper from overstating the case. The honest scope of the examples is a real strength.\n\nThe main soft spot is exactly where the stress-test note puts its finger. The abstract and introduction claim Gibbs phenomena are eliminated in the limit q→1 on certain meshes, but several theorems only prove existence of an L1-best approximation without over/undershoot. For Theorem 1.4(4), the proof says the limit of odd Lq minimizers must be odd and therefore must be the odd L1 minimizer. That skips compactness and continuity; in finite dimensions a cluster-point argument can repair it, because every cluster point is odd and the odd L1 minimizer is unique. But the repair does not extend to the N-element and 2D cases (Theorems 1.1(2,3), 1.5(4,5)): the L1 minimizer is not unique there, and nothing rules out other L1 minimizers with overshoot as cluster points of the Lq sequence. The numerical evidence in Section 6.3.1 is suggestive, but it is not a proof. So the central advertised claim is underproved in the cases that matter most for the design heuristic.\n\nThere are also minor blemishes: no code for the FEniCS experiments, a few typos, and some loose language calling an L1-best approximation unique when a family exists. None of this sinks the paper.\n\nWho is this for? Researchers working on L1-based or minimum-residual FEM, and anyone interested in the approximation-theoretic side of Gibbs phenomena. It deserves a serious referee. I would send it out, with a request that the authors either add the missing convergence arguments where they can, or reframe the claims as 'there exists an L1-best approximation without overshoot' and reserve the q→1 convergence claim for the cases where it is actually proven.","headline":"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.","tokens_in":31526,"tokens_out":3958,"would_cite":true,"duration_ms":295108,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["41A50","65N30"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Lq-best approximation","Gibbs phenomenon","finite element spaces","piecewise linear approximation","L1 minimization","mesh grading","non-uniqueness of best approximation"],"falsifier":"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 $u_h(h)\\to1$.","tokens_in":30525,"feed_emoji":"📐","tokens_out":9817,"duration_ms":87381,"temperature":0.7,"pith_summary":"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.","feed_headline":"Gibbs overshoot vanishes as q→1 on meshes built to avoid it","feed_subtitle":"For piecewise-linear finite element fits, the overshoot disappears only when elements near the jump are not too small.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $L^2$."],"supporting_citations":[{"why":"Establishes the one-dimensional phenomenon this paper extends: for best $L^p$ approximation of a jump by polygonal lines, Gibbs oscillations vanish as $q\\to1$.","marker":"Saff and Tashev (1999)"},{"why":"Supplies the subdifferential characterization of best approximation used to derive every necessary condition in the paper.","marker":"Singer (1970)"},{"why":"Gives the explicit form of the subdifferential of the norm used in the $L^1$ and $L^q$ characterisations.","marker":"Cioranescu (1990)"},{"why":"Documents the Gibbs phenomenon in $L^1$ trigonometric approximation, providing the context that $L^1$ best approximation can still oscillate.","marker":"Moskona, Petrushev, and Saff (1995)"},{"why":"Motivates the search for finite element methods in $L^1$ by showing that first-order PDEs are naturally posed there.","marker":"Guermond (2004)"},{"why":"Applies the mesh-design idea to a nonlinear Petrov-Galerkin method for convection-diffusion-reaction, the stated practical target of the paper.","marker":"Houston et al. (2019)"}],"fun_headline_variants":["Gibbs effect dies as q→1 on the right mesh","For L1 fits, mesh shape decides Gibbs overshoot","q=1 best fits kill Gibbs on graded meshes","No Gibbs at q=1: mesh geometry is the key"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Gibbs effect dies as q→1 on the right mesh","For L1 fits, mesh shape decides Gibbs overshoot","q=1 best fits kill Gibbs on graded meshes","No Gibbs at q=1: mesh geometry is the key"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000182,"raw_usage":{"total_tokens":1366,"prompt_tokens":1059,"completion_tokens":307,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":675,"completion_tokens_details":{"reasoning_tokens":238}},"tokens_in":675,"tokens_out":307,"duration_ms":3576,"temperature":1.0,"reasoning_tokens":238,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T05:41:05.272401+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $u_h(h)\\to1$.","supporting_citations":[],"review_version":1}