{"id":"e45452e5-07b8-451a-988c-358780c404dc","arxiv_id":"1908.03087","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A mixed face-centred finite volume method with a linear in-cell solution and constant face unknowns reaches second-order convergence without increasing the global system size relative to the first-order scheme.","lead":"This paper proposes a second-order face-centred finite volume method that stores the solution on mesh faces and eliminates interior degrees of freedom cell by cell. The method achieves second-order accuracy for Poisson and Stokes problems with a global system the same size as the cheaper first-order version, and it appears robust on badly distorted and stretched meshes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Second-order claim hinges on an unverified inheritance from reduced-stabilisation HDG; the constant face space may fall outside the cited theory.","rationale":"The central algorithmic claim, that the global system has identical size and sparsity to the first-order FCFV, is structurally true and not in dispute. The second-order convergence claim is strongly supported by the numerical experiments: optimal rates are observed in 2D and 3D, for Poisson and Stokes, and on distorted and stretched meshes. The reader's conditional verdict is therefore reasonable. The weakest link is the theoretical justification: the paper invokes Oikawa's reduced-stabilisation HDG analysis, but the FCFV spaces (linear volumetric u, constant volumetric q, constant face uhat) are not the standard HDG spaces, since the face variable is a proper subspace of the trace of the volumetric space. The paper does not show that Oikawa's theorems apply to this nonconforming setting, and it does not provide an independent convergence or inf-sup proof. This is exactly the concern identified by the reader. A literature check or a derivation of the Crouzeix-Raviart equivalence would settle it. Since the numerical evidence is strong and the missing proof does not contradict any internal inconsistency, the appropriate verdict remains CONDITIONAL, i.e., unchanged.","tokens_in":15515,"tokens_out":24920,"duration_ms":280529,"concrete_test":"Obtain the precise hypotheses of the convergence and inf-sup theorems in Oikawa (2015, J Sci Comput 65 and 2016, J Sci Comput 67) and check them against the FCFV spaces (P1/P0/P0). If the theorems require uhat in the trace space of u (P1 per face), the inheritance argument is invalid; then settle the claim by proving equivalence of the Poisson method to the nonconforming Crouzeix-Raviart method on the same meshes (which would supply second-order u and first-order q), and for Stokes by verifying the discrete inf-sup condition numerically on the distorted/stretched meshes of Section 5.5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim of second-order convergence is supported by numerical experiments, but its theoretical justification is an inheritance argument: Section 1 and Remark 2 state that the method 'inherits the convergence properties of HDG' from Oikawa's reduced-stabilisation HDG [24,25]. The FCFV discretisation uses u in P1 per cell, q in P0 per cell, and uhat in P0 per face. This combination is not the standard HDG setting: the face space is a proper subspace of the trace of the element space (P1), so the method is nonconforming. The paper does not verify that the hypotheses of the theorems in [24,25] permit this reduced face space, nor does it prove LBB stability for the Stokes saddle-point system (41). If the cited analysis does not cover this case, the second-order and inf-sup claims rest on the numerical evidence alone, which is extensive but not a proof. The missing step is not merely cosmetic: the whole point of the method is that the constant face space is what keeps the global system identical to the first-order FCFV, so the convergence theory must be specific to this reduced face space.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a second-order face-centred finite volume (FCFV) method for Poisson and Stokes problems. The method uses a mixed formulation with a linear approximation of the solution inside each cell, a constant approximation of its gradient, and a constant approximation of the solution on each cell face. A key ingredient is a projection onto constants in the numerical flux, which the authors state is necessary for second-order accuracy. The solution on the faces is the only globally coupled unknown, so the global system has the same size and sparsity as the previously proposed first-order FCFV method. The paper presents numerical experiments in two and three dimensions showing second-order convergence for the solution/velocity, first-order convergence for the gradient, robustness on distorted and stretched meshes, CPU-time comparisons with the first-order FCFV, two large-scale three-dimensional examples, and an h-adaptive strategy based on the difference between the first- and second-order schemes.","tokens_in":15683,"tokens_out":6486,"duration_ms":71816,"significance":"If the convergence and stability claims are correct, the method is a practically attractive way to obtain second-order accuracy for elliptic and Stokes problems without gradient reconstruction, while keeping a global system of the same size as a first-order method. The numerical evidence is extensive: convergence studies in 2D and 3D, distorted and stretched meshes, CPU-time comparisons, a problem with over five million cells, and an adaptive strategy tested on a complex geometry. The paper is also commendable for reporting timings and for reusing computations between the two schemes in the error indicator. The main weakness is that the central second-order and LBB claims are not proved in the paper but are asserted to be inherited from cited reduced-stabilisation HDG analyses, without verifying that the specific reduced face space used here satisfies the hypotheses of those analyses.","major_comments":[{"comment":"The central claim of second-order convergence is justified only by an inheritance argument: the paper states that the FCFV method 'inherits the convergence properties of HDG' from Oikawa [24,25]. However, the discretisation uses u in V1, q in V0, and the trace unknown in the constant face space V̂0, which is a proper subspace of the trace of V1. The paper does not identify the precise theorems in [24,25], nor does it verify that their hypotheses (in particular on the face approximation space and the role of the projection P0 in the numerical flux) are satisfied by this P1/P0/P0 combination. Because the constant face space is exactly what makes the global system identical in size to the first-order FCFV, the convergence theory must be specific to this reduced face space. The authors should either provide a proof or state and verify the relevant theorem from [24,25] with all hypotheses checked.","section":"Sections 1 and 2.3, Remark 2"},{"comment":"For the Stokes problem, the paper asserts that the method 'passes the LBB condition' and presents the discrete problem as the saddle-point system (41), but no inf-sup analysis or proof is given. The well-posedness of the reduced-order Stokes saddle-point problem is load-bearing for the numerical results in Section 5.2 and for the adaptive examples. The authors should provide a discrete inf-sup proof for the constant face space, or precisely identify the result in [25] that covers this case and explain why its assumptions hold here.","section":"Section 3, Eq. (41) and Concluding Remarks"}],"minor_comments":[{"comment":"The relation between the a priori estimate (46) and the desired mesh size (47) is unclear: if ε_e ≤ C h^{1+n_sd/2}, the exponent in (47) should be 1/(1+n_sd/2), not the expression as printed. Please clarify the formula.","section":"Section 4, Eq. (47)"},{"comment":"There are several typographical errors: 'veloocity gradient' in Section 5.3 and 'randomnly disorted' in Section 5.5. These should be corrected.","section":"Section 5.3 and 5.5"},{"comment":"The operator P0 is introduced as 'the projection operator over the space of constant functions', but it is used on each face individually. It would be clearer to define P0 explicitly as the L2 projection onto constants on each face Γ_e,i, as is done in the proof of Lemma 1.","section":"Section 2.3, Eq. (10)"},{"comment":"The claim in the abstract that the method requires a global system with 'identical size and identical number of non-zero elements' is accurate for the global matrix, but the total cost is not identical since the local problems are larger and the assembly time is higher, as noted in Section 6.1. The wording could be qualified to avoid overstatement.","section":"Section 6.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a solid numerical methods paper with convincing experiments, but the theoretical justification for the flagship second-order claim is a citation to work that may or may not cover the exact nonconforming reduced face space used here. Since the claim is the main selling point and the same structure is carried over to Stokes, I would ask the authors to close this gap with either a proof or a precise, verified inheritance statement before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a genuinely useful extension of the authors' FCFV method, and the headline claim checks out. By moving the cell unknown from P0 to P1 while keeping the face trace in P0, they get second-order convergence for the solution and first-order for the gradient with the same global matrix size as the first-order method. The stress-test concern about Oikawa's theory does not land, in my view. The P0 face space is not an ad hoc nonconforming choice; it is the k=0 instance of the reduced-stabilisation HDG setting in [24,25], where the trace space is P_k and the stabilization projects the higher-degree element trace down to that space. So the inheritance argument in Section 1 and Remark 2 is legitimate, not hand-waving. The paper would be clearer if it said this mapping explicitly rather than just asserting that the method 'inherits' convergence properties, but the cited analysis does cover the setting.\n\nWhat is actually new here is the second-order FCFV formulation for Poisson and Stokes, the projection-based numerical flux that makes it work, the cheap error indicator obtained by comparing the projected and unprojected versions, and a thorough numerical study. The mesh-distortion and stretching results are convincing, and the aircraft and sphere examples show the method runs at real scale. I also believe the CPU-time comparisons against their own first-order method; the global system is indeed the same size, and the local 3x3 or 4x4 solves are cheap. No code or data is shipped, so the numerical evidence is not independently reproducible from the paper, which is a real but minor gap.\n\nSoft spots are proportionate. The convergence and inf-sup results are not proved in the paper; they are imported from Oikawa. That is acceptable if the import is correct, and I think it is, but the paper should pin down the exact hypotheses, especially for Stokes and for the strongly stretched-mesh regime. The experimental comparison is only against FCFV-1; adding a mature second-order FV or standard HDG solver would make the advantage claims sharper. The stabilization parameter is chosen from numerical experiments (tau >= 100 gives a plateau), not derived, though the insensitivity study partly addresses this. The adaptive indicator is an indicator, not a certified estimator; the efficiency plots are encouraging but not a proof.\n\nBottom line: this is a solid methods paper for people working on HDG-like finite volumes and large-scale elliptic and Stokes solvers. It deserves serious peer review, and I would read it carefully rather than desk-reject.","headline":"A solid, practically useful second-order extension of FCFV whose theoretical inheritance from Oikawa's reduced-stabilisation HDG is legitimate, though the paper would benefit from a sharper statement of that link and an independent baseline.","tokens_in":16224,"tokens_out":7261,"would_cite":true,"duration_ms":79915,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65N08","65N30","65N12"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims a second-order face-centred finite volume method for elliptic problems that solves a global system identical in size and sparsity to the first-order method.","keywords":["finite volume method","face-centred","second-order convergence","hybridisable discontinuous Galerkin","Poisson equation","Stokes flow","mesh adaptivity","elliptic problems"],"falsifier":"Solve the Poisson problem on a sequence of tetrahedral meshes with stretching factor 1000 as in Section 5.5, and compute the relative $L^2$ error of the solution against the exact solution. If the measured convergence order is not close to 2 (or if it degrades with mesh distortion), the paper's central claim of second-order accuracy at a first-order global cost fails; the paper's own Figures 10 and 12 are the evidence it would need to match.","tokens_in":15304,"feed_emoji":"🧮","tokens_out":8095,"duration_ms":79460,"temperature":0.7,"pith_summary":"This paper aims to show that a finite volume method can be made second-order accurate for elliptic problems without paying the usual price. The method stores the unknown on mesh faces rather than cell centres or vertices, solves a small local problem in each cell, and then solves one global system whose only unknowns are the face values. The central claim is that this global system has exactly the same size and sparsity pattern as the previously proposed first-order face-centred method, yet the solution converges at second order in the mesh size while the gradient converges at first order. The authors further claim the method is insensitive to mesh distortion and stretching, and that the difference between the first- and second-order variants supplies a built-in error indicator for adaptive refinement.","feed_headline":"Second-order accuracy from a first-order-sized global solve","feed_subtitle":"Storing unknowns on mesh faces doubles solution accuracy without enlarging the global system, even on distorted meshes.","key_machinery":"The key mechanism is a projection operator $P_0$ placed inside the numerical flux: on each face the flux is $n\\cdot q_h + \\tau_e(P_0 u_h - \\hat u)$ (with Dirichlet data in place of $\\hat u$ on the Dirichlet boundary). $P_0$ sends the linear in-cell solution to its constant average on the face, and it is the single ingredient that turns the first-order face-centred method into a second-order one. The paper proves in Lemma 1 that $\\int_{\\Gamma_{e,i}} P_0 u_h\\,d\\Gamma = \\int_{\\Gamma_{e,i}} u_h\\,d\\Gamma$, so the projection only changes the cell-level matrix $m_e$ and leaves the global system's size and sparsity untouched. The rest of the machinery is the standard HDG/FCFV split: a local Dirichlet problem per element eliminates the interior solution and gradient in terms of the face unknowns, and the global problem enforces flux continuity on the faces.","core_discovery":"The paper claims that a face-centred finite volume method, which stores the unknown on mesh faces and treats the gradient as an independent variable, can be upgraded from first to second order convergence in the solution without enlarging the globally coupled system. The construction follows hybridised discontinuous Galerkin methods with reduced stabilisation: inside each cell the solution is approximated by a linear polynomial, while the gradient and the face unknowns are piecewise constant, and the numerical flux contains a projection of the cell solution onto constants. Removing that projection collapses the method back to first order, which is precisely why the difference between the two variants can be used as an error indicator. The global system, whose unknowns are the face values, is claimed to have the same size and the same number of non-zero entries as the first-order method because the face space is unchanged. Numerical experiments show second-order convergence for the solution, first-order convergence for the gradient, the same rates on distorted and stretched meshes, and for Stokes flow the method passes the LBB condition and delivers second-order velocity and first-order pressure convergence.","pith_inferences":["The same device should extend to higher-order face-centred methods: using degree-$k$ polynomials inside cells with a projection onto the trace space would likely reproduce the HDG convergence hierarchy while keeping the face unknowns fixed, so the global cost would remain tied to the face space rather than the interior order.","The cheap error indicator suggests an industrial-grade adaptive strategy: run the first- and second-order variants simultaneously, use their difference to mark cells, and re-mesh only where that difference is large; the paper's corrugated-channel example is a step in that direction.","If the claim of distortion insensitivity holds for strongly deformed cells, face-centred volume methods could be a natural fit for Lagrangian and moving-mesh simulations, where classical cell-centred reconstructions degrade.","A missing piece, left implicit by the paper, is a full convergence proof for this projection-based FCFV on the distorted meshes used in Section 5.5; numerical evidence is strong, but the theoretical transfer from the cited reduced-stabilisation analysis to this setting is assumed rather than shown."],"forward_implications":["For a given mesh, the solution error drops from first to second order with no extra global unknowns, so a target accuracy is reached with far fewer degrees of freedom.","Because the gradient is a direct unknown, no reconstruction step is needed, and the method keeps its convergence rates on strongly distorted and stretched meshes.","The global linear system has the same size and non-zero pattern as the first-order method, so the extra accuracy is obtained at essentially the same assembly and solve cost per mesh.","The difference between the first- and second-order solutions gives a computable error indicator that drives h-adaptivity without solving an auxiliary problem.","For Stokes flows, the method preserves the LBB condition and gives second-order velocity, first-order pressure and gradient; the paper demonstrates this in problems with over 5 million tetrahedra."],"supporting_citations":[{"why":"Supplies the reduced-stabilisation hybridised discontinuous Galerkin analysis that the method builds on to achieve second-order accuracy through the projection operator.","marker":"[24]"},{"why":"Extends that analysis to the Stokes problem, underpinning the LBB condition and the convergence claims for the Stokes variant.","marker":"[25]"},{"why":"Defines the original first-order face-centred finite volume method that this work upgrades and serves as the baseline for cost and accuracy comparisons.","marker":"[28]"},{"why":"Provides the unified hybridization framework that justifies the element-by-element local solve followed by a face-only global system.","marker":"[8]"},{"why":"Gives the velocity-pressure HDG formulation for Stokes used here, including the pressure compatibility condition and the local-global split.","marker":"[20]"}],"fun_headline_variants":["Second-order accuracy from a first-order-sized solve","Face-centred FV goes second-order with same global matrix","Double accuracy without extra global unknowns","Face-based method hits second order at first-order cost","Second-order FV without bigger linear system"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper relies on the assumption that the mathematical analysis of a related, more general family of methods carries over to this particular face-centred scheme, including on highly distorted and stretched meshes; it does not prove that carry-over itself.","fun_headline_variants_meta":{"raw":{"variants":["Second-order accuracy from a first-order-sized solve","Face-centred FV goes second-order with same global matrix","Double accuracy without extra global unknowns","Face-based method hits second order at first-order cost","Second-order FV without bigger linear system"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000182,"raw_usage":{"total_tokens":1309,"prompt_tokens":945,"completion_tokens":364,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":561,"completion_tokens_details":{"reasoning_tokens":294}},"tokens_in":561,"tokens_out":364,"duration_ms":4129,"temperature":1.0,"reasoning_tokens":294,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:25:06.396267+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve the Poisson problem on a sequence of tetrahedral meshes with stretching factor 1000 as in Section 5.5, and compute the relative $L^2$ error of the solution against the exact solution. If the measured convergence order is not close to 2 (or if it degrades with mesh distortion), the paper's central claim of second-order accuracy at a first-order global cost fails; the paper's own Figures 10 and 12 are the evidence it would need to match.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the reduced-stabilisation hybridised discontinuous Galerkin analysis that the method builds on to achieve second-order accuracy through the projection operator."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Extends that analysis to the Stokes problem, underpinning the LBB condition and the convergence claims for the Stokes variant."},{"cited_title":"Sevilla, M","cited_arxiv_id":null,"evidence_quote":"Defines the original first-order face-centred finite volume method that this work upgrades and serves as the baseline for cost and accuracy comparisons."},{"cited_title":"Cockburn, J","cited_arxiv_id":null,"evidence_quote":"Provides the unified hybridization framework that justifies the element-by-element local solve followed by a face-only global system."},{"cited_title":"Nguyen, J","cited_arxiv_id":null,"evidence_quote":"Gives the velocity-pressure HDG formulation for Stokes used here, including the pressure compatibility condition and the local-global split."}],"review_version":1}