{"id":"7e00edd5-5990-4ec0-865c-a5acf3a6e66e","arxiv_id":"2509.06480","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A three-step sequential decoupled DG scheme for nonlinear thermo-poroelasticity is stable and converges at rate tau plus optimal spatial rates.","lead":"This paper proposes a sequential decoupling scheme for coupled heat, fluid pressure and rock deformation, where the three unknown fields are solved one after another at each time step using a discontinuous Galerkin method. The authors prove stability and optimal convergence and report that the scheme is several times faster than a fully implicit solver.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Optimal h^{k2+1} pressure error in Corollary 4.1 relies on an integration-by-parts identity that is invalid for discontinuous Galerkin functions: face and boundary terms are dropped without control.","rationale":"The reader's weakest assumption concerns the cut-off constant M and the time-step restriction τ ≤ C3(a0−b0)/(4M^2). I do not regard this as the most load-bearing issue: M is fixed by the a priori flux bound and the restriction is a standard small-time-step condition, albeit with an unquantified constant. The more serious problem is in the proof of Theorem 4.1 itself. The estimate of I10 requires an elementwise integration by parts of a term involving ρ_p and θ_T, both discontinuous. The proof drops the resulting boundary and interior-face terms as if the functions were H^1 with zero traces. This is internally inconsistent with the DG setting: the projections (3.6)-(3.7) do not enforce zero trace, and the discrete spaces are broken. Without a control of these face contributions, the O(h^{2k2+2}) bound for H1, and hence the h^{k2+1} pressure convergence in Corollary 4.1, is unsupported. The paper has genuine strengths: the numerical experiments confirm the advertised rates, the stability analysis is substantial, and most of the surrounding estimates are standard. But this proof gap affects the central theoretical claim and should be repaired or acknowledged. Since the gap is localized and likely fixable with a more careful face-term analysis, I would keep the reader's CONDITIONAL verdict rather than escalate to rejection.","tokens_in":29840,"tokens_out":15086,"duration_ms":165360,"concrete_test":"Check the missing face terms on the §5 example with k1=k2=k3=1, p=T=sin(πx)sin(πy), and the projections (3.6)-(3.7). Compute S_h = Σ_{e∈Γ_I}∫_e (∇T·n_e)(ρ_p^+ θ_T^+ − ρ_p^- θ_T^-) ds + ∫_{∂Ω} ρ_p (∇T·n) θ_T ds on uniform meshes. If |S_h| decays like h^3 rather than h^4 as h→0, the dropped term is one power of h worse than the I10 bound used in Theorem 4.1, so Corollary 4.1's h^{k2+1} pressure estimate is not established by the proof. If the authors can show S_h is absorbed by the c-projection/coercivity or is O(h^4), the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central estimate Corollary 4.1 inherits its h^{k2+1} pressure order from the bound on I10 in Theorem 4.1. In the treatment of H1 = (K∇ρ_p^{n+1}·∇R_T T^{n+1}, θ_T^{n+1}), the proof integrates by parts and then discards the boundary term ⟨ρ_p^{n+1}, ∇T^{n+1}·θ_T^{n+1} n⟩_{∂Ω}, citing the boundary condition. But ρ_p = p − R_p p and θ_T = R_T T − T_h are DG functions: R_p is not trace-zero, θ_T does not vanish on ∂Ω, and on internal edges the summation by parts produces the jump term Σ_{e∈Γ_I}∫_e (∇T·n_e)(ρ_p^+ θ_T^+ − ρ_p^- θ_T^-) ds. These face terms are neither zero nor estimated anywhere in the paper. The claimed bound |H1| ≤ C(h^{2k2+2}+h^{2k3+2}) is exactly what improves the pressure error from h^{k2} to h^{k2+1}; a direct estimate only gives h^{2k2}. Thus the advertised optimal spatial order for p has a missing face-term estimate. This is independent of the cut-off M caveat: M can in principle be fixed after a priori flux bounds, whereas the face terms are present for every M.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a sequential (non-iterative) splitting method for the fully coupled quasi-static thermo-poroelasticity system with nonlinear convective transport. The spatial discretization is a symmetric interior penalty discontinuous Galerkin method and the temporal discretization is backward Euler. At each time step the pressure, temperature, and displacement are solved successively using previously computed values, with a cut-off operator applied to the convective Darcy-flux term. The authors prove existence and uniqueness of the discrete solution, a stability estimate under a time-step restriction, and an error estimate of order O(τ + h^{k1} + h^{k2+1} + h^{k3+1}) when the first-step values are obtained by the fully implicit scheme of their earlier paper. Numerical experiments with P1 elements report the expected first-order time convergence and second-order spatial convergence for pressure and temperature, and show a CPU-time advantage over the fully implicit DG method.","tokens_in":30279,"tokens_out":8321,"duration_ms":98128,"significance":"Sequential decoupling methods for multiphysics problems are practically attractive because they avoid inner iterations and reduce the size of the algebraic systems. If the advertised error estimate is correct, this would be a worthwhile contribution: it extends the sequential DG idea from Biot poroelasticity to nonlinear thermo-poroelasticity and, for the first time for this problem class, provides a stability analysis and optimal space-time error estimates. The cut-off treatment of the convective term and the two initialization strategies are also useful for practitioners. The numerical study is careful and reports rates that match the stated orders. However, the central convergence proof contains a load-bearing gap in the treatment of the convective consistency term; until that gap is closed, the optimal pressure-order claim is not established.","major_comments":[{"comment":"The proof of |H1| integrates (K∇ρ_p^{n+1}·∇R_TT^{n+1}, θ_T^{n+1}) by parts and then discards the boundary term ⟨ρ_p^{n+1}, ∇T^{n+1}·θ_T^{n+1}n⟩_{∂Ω}, citing the homogeneous boundary condition. But ρ_p^{n+1}=p^{n+1}-R_pp^{n+1} and θ_T^{n+1}=R_TT^{n+1}-T_h^{n+1} are discontinuous Galerkin functions: R_p does not have zero trace and θ_T does not vanish on ∂Ω. Moreover, summing by parts over elements produces internal-face terms Σ_{e∈Γ_I}∫_e (∇T·n_e)(ρ_p^+θ_T^+−ρ_p^-θ_T^-)ds. These face terms are neither zero nor estimated anywhere. A direct trace estimate gives only O(h^{k2}) for the pressure error, i.e. one power less than the advertised h^{k2+1} in Corollary 4.1. Since the H1 bound is what raises the pressure order from h^{k2} to h^{k2+1}, the optimal spatial estimate is not established as written.","section":"§4.2, estimate of H1 after (4.12)"},{"comment":"The estimate of (∇ρ_T^{n+1}·∇ρ_p^{n+1}, θ_T^{n+1}) uses the bound ||∇ρ_T^{n+1}||_{0,∞} ≤ C h^{k3} ||T||_{k3+1,∞}, citing [22]. Assumption A6 only provides T∈L∞(0,tf;H^{k3+1}) and ∂_tT∈L∞(0,tf;W^{1,l}) with l>d; it does not imply T∈W^{k3+1,∞}. Thus the L∞ projection estimate used in the proof is not available under the stated assumptions. Either Assumption A6 must be strengthened, or this term needs a different estimate that is valid under the current regularity assumptions.","section":"§4.2, H1 bound and Assumption A6"},{"comment":"The well-posedness and stability conditions are τ ≤ 2C_3a_0/M^2 and τ ≤ C_3(a_0−b_0)/(4M^2), respectively, where M is the cut-off constant. The paper only specifies that M is 'large enough' so that M(K∇p^n)=K∇p^n for the exact flux; the precise value is never quantified. Since M depends on the a priori sup norm of the exact Darcy flux, the allowable time step is tied to an unquantified problem-dependent constant. If the exact flux is large, the guaranteed stability time step may be prohibitively small. The authors should state explicitly that M is chosen from the a priori bound and discuss the resulting time-step restriction, or reformulate the stability analysis to avoid this dependence.","section":"Theorems 3.1 and 3.2"}],"minor_comments":[{"comment":"The statement says '(u^n_h,p^n_h,T^n_h) be the solutions of (4.5)-(4.7) and (3.2)-(3.4)', but (4.5)-(4.7) are consistency identities for the exact solution. The exact solution should be for (2.1)-(2.7).","section":"Theorem 4.1 statement"},{"comment":"The paper presents Option 1 as a practical way to start the scheme, and Table 9 tests it, but no error estimate is proved for the Option 1 initialization (3.8)-(3.10). Corollary 4.1 only covers Option 2. If Option 1 is meant to be a rigorous alternative, an analysis or a clear statement that it is heuristic should be added.","section":"Remark 3.1 and Option 1"},{"comment":"The bound for H2 uses (4.2), but (4.2) is for ||p^{n+1}-p^n||; the estimate needed is for ∇(R_pp^{n+1}-R_pp^n). A gradient version should be stated or derived.","section":"§4.2, H2 estimate"},{"comment":"The symbol H1 is used both for a term in (4.12) and for the Sobolev space H^1. Renaming one of them would avoid confusion.","section":"Notation"},{"comment":"The spatial convergence tables stop at h=1/32. Since the observed pressure rates are close to 2 but not fully asymptotic, one or two finer levels would strengthen the numerical confirmation, although the present rates are already consistent with the claimed orders.","section":"Section 5"}],"recommendation":"major_revision","confidential_remarks":"The decisive issue is the missing face-term estimate in the H1 bound of Theorem 4.1. The paper's main advertised optimal pressure order depends on that estimate. The initialization estimate is imported from the authors' own prior work [18]; this is acceptable because [18] is published, but the dependency should be stated more prominently. If the authors can close the H1 gap without losing an order, I would support publication; otherwise the theorem should be weakened accordingly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague —\n\nThe headline is that the paper claims an optimal h^{k2+1} pressure error estimate for a sequential DG method, but that estimate has a proof gap. In Theorem 4.1, the bound on the term H1 = (K∇ρ_p^{n+1}·∇R_T T^{n+1}, θ_T^{n+1}) drops all face terms after integration by parts, citing the boundary condition. But ρ_p = p − R_p p and θ_T = R_T T − T_h are DG functions; R_p is not trace-zero, and interior edges produce jump terms that are neither zero nor estimated. Without those face-term estimates, the direct bound only gives h^{2k2}, not the h^{2k2+2} that powers the optimal pressure order. This is independent of the cut-off M caveat — the face terms are there for any M.\n\nThat said, the paper is not a throwaway. It builds a three-step sequential SIPG scheme for nonlinear thermo-poroelasticity with no internal iterations, proves existence, uniqueness, and a stability estimate, and runs systematic experiments across coupling strengths with observed first-order time and optimal spatial rates. The efficiency comparison against the fully implicit DG method shows a real speedup. That is a genuine within-subfield contribution, and the numerical work is reproducible in spirit, though no code is shipped.\n\nThe other soft spots are lesser. The initialization Option 1 relies on an admitted tiny τ0 heuristic; Option 2 borrows the fully implicit start from the authors' earlier paper [18], which is prior published work and not a circularity. The cut-off constant M is never quantified, so the stability condition τ ≤ C3(a0−b0)/4M^2 is not checkable a priori; the paper hand-waves a bounded-flux assumption. That is a real but secondary concern.\n\nWho is this for? Numerical analysts in poromechanics or splitting methods. It deserves a serious referee — the methodology is sound enough and the claim important enough — but the referee should ask the authors to supply the missing face-term estimates or retract the optimal pressure order. As it stands, Corollary 4.1 is not proven.\n\nI would not cite the convergence rate in my own work until the gap is closed. Bring it to a reading group if you want a live example of DG proofs going astray on boundary terms.","headline":"A useful sequential DG scheme with an unproven optimal-order claim: the pressure estimate drops DG face terms, so the advertised h^{k2+1} order rests on a gap.","tokens_in":30680,"tokens_out":4773,"would_cite":false,"duration_ms":48951,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M15","65M60","76S99"],"pacs":[],"model":"deepseek-v4-flash","headline":"A three-step sequential DG scheme for fully coupled thermo-poroelasticity is stable and achieves optimal convergence in time and space, provided the first step comes from a fully implicit solve.","keywords":["thermo-poroelasticity","sequential decoupling","discontinuous Galerkin","symmetric interior penalty","backward Euler","cut-off operator","optimal convergence","stability analysis"],"falsifier":"Take a manufactured solution with a Darcy flux strong enough that the exact flux exceeds the cut-off M used in the code. If the scheme keeps optimal order while the condition M(K∇p_h)=K∇p_h is violated on the exact flux, the cut-off requirement is not actually load-bearing; if it loses order or becomes unstable, the unquantified choice of M is the controlling assumption. Compare with the fully implicit DG method of [18] at the same time step to separate splitting error from cut-off error.","tokens_in":1743,"feed_emoji":"🧮","tokens_out":2036,"duration_ms":87558,"temperature":0.7,"pith_summary":"The paper proposes a way to solve the fully coupled quasi-static thermo-poroelasticity system with nonlinear convective transport without solving the three physical fields together: at each time step it solves pressure from the flow equation, then temperature from the heat equation using that pressure, then displacement from the mechanical equation using both. The claim is that this one-way splitting, discretized by symmetric interior penalty discontinuous Galerkin in space and backward Euler in time, is well-posed and stable, and that its error obeys the optimal bound C(τ + h^{k1} + h^{k2+1} + h^{k3+1}). The significance is practical: sequential solves avoid internal iterations and reduce the size of linear systems, while the proven error matches the fully implicit method. The analysis leans on a cut-off operator that truncates the Darcy flux in the convection term, and on a fully implicit first step to get optimal initial values.","feed_headline":"Sequential splitting hits optimal error without inner iterations","feed_subtitle":"Pressure, then temperature, then displacement: three one-way DG solves per step, stable and first-order in time.","key_machinery":"The mechanism that makes the proof work is the cut-off operator M, defined as the identity below a level M and as radial truncation above it, applied to the Darcy flux in the convective term (M(K∇p_h^n)·∇T_h^{n+1}). It renders the convection operator Lipschitz and L∞-bounded by M, giving existence, uniqueness, and stability under the time-step bound τ ≤ C3(a0-b0)/(4M^2); the sequential ordering pressure→temperature→displacement removes the need for internal iterations, and a large SIP penalty parameter plus the stabilization γ stabilize the DG bilinear forms.","core_discovery":"The central result is Corollary 4.1: under Assumption A on regularity of coefficients and solution, the numerical solution of the sequential scheme (3.2)-(3.4) satisfies |||u^n - u_h^n|||_V + ||p^n - p_h^n|| + ||T^n - T_h^n|| ≤ C(τ + h^{k1} + h^{k2+1} + h^{k3+1}), where k1, k2, k3 are the polynomial degrees for displacement, pressure, and temperature. This is presented as the first stability analysis for this type of sequential discontinuous Galerkin splitting for the fully coupled nonlinear thermo-poroelasticity model, and the first to obtain optimal first-order convergence in time and optimal polynomial orders in space. The theorem requires the first-layer values (p_h^1, T_h^1, u_h^1) to c","pith_inferences":["Editorial inference: the unquantified cut-off level M means the stability condition τ ≤ C3(a0-b0)/(4M^2) cannot be checked ahead of time; a practical refinement would estimate M from the computed flux and verify the constraint a posteriori.","Editorial inference: the paper's experiments stay in mild parameter regimes; a natural stress test is a convection-dominated case where |K∇p_h| approaches M, to see whether the time-step restriction or the truncation degrades accuracy.","Editorial inference: because the error estimate is built on the regularity Assumption A6, testing rough coefficients or lower-regularity solutions would reveal whether that assumption is necessary in practice or merely a proof convenience.","Editorial inference: the sequential ordering admits an operator-splitting interpretation, so viewing the cut-off as a flux limiter could connect this analysis to splitting-error and positivity-preserving approaches for nonlinear transport in porous media."],"forward_implications":["If the error estimate is correct, users get optimal accuracy without ever forming the coupled monolithic system: each time step costs three smaller linear solves instead of one large coupled one.","The fully implicit first step is not just an implementation detail; dropping it forces a very small initial time step and an extra assumption that the displacement increment is negligible, as in Option 1.","The time-step restriction depends on M, so the practical range of τ is tied to the strength of the Darcy flux through the cut-off level.","The reported experiments confirm first-order time convergence and polynomial-degree spatial convergence across five different coupling strengths, and compare favorably in CPU time with the fully implicit DG scheme of [18].","The same sequential pattern with a cut-off-stabilized convective term could be applied to other DG discretizations of coupled flow-mechanics problems."],"supporting_citations":[{"why":"Provides the sequential discontinuous Galerkin splitting idea for the Biot model that the paper extends to the three-field thermo-poroelasticity system.","marker":"[15]"},{"why":"Supplies the fully implicit SIPDG/backward-Euler scheme used for optimal first-layer values and the consistency lemma underpinning the error equations.","marker":"[18]"},{"why":"Introduces the cut-off operator M and splitting solution schemes for the same nonlinear thermo-poroelasticity model; the paper adapts M to a non-iterative sequential scheme.","marker":"[13]"},{"why":"Establishes well-posedness and the regularity Assumption A for the quasi-static thermo-poroelastic equations, the backdrop for the discrete analysis.","marker":"[14]"},{"why":"Provides the uniform Lipschitz property of the cut-off operator used in existence, uniqueness, and stability estimates.","marker":"[42]"},{"why":"Supplies the trace and inverse inequalities, elliptic projection estimates, and DG bilinear-form coercivity used throughout the error analysis.","marker":"[39]"},{"why":"Supplies the interior penalty DG coercivity and norm equivalence used for the mechanical and diffusion bilinear forms.","marker":"[5]"},{"why":"Supplies the Korn-type inequality used to control the displacement gradient by the DG energy norm.","marker":"[26]"}],"fun_headline_variants":["Optimal error for thermo-poroelasticity without inner iterations","Three one-way DG solves per step: optimal error proven","First stability proof for sequential DG in poroelasticity","No inner loops: sequential DG hits optimal convergence","One-way splitting solves thermo-poroelasticity accurately"],"cache_read_input_tokens":32384,"weakest_assumption_plain":"The whole stability and error proof requires one cut-off constant M that is large enough to leave the exact Darcy flux unchanged and small enough that the time step satisfies τ ≤ C3(a0-b0)/(4M^2); since M is never quantified and depends on an a priori bound on the true flux, the resulting time-step restriction cannot be verified in practice.","fun_headline_variants_meta":{"raw":{"variants":["Optimal error for thermo-poroelasticity without inner iterations","Three one-way DG solves per step: optimal error proven","First stability proof for sequential DG in poroelasticity","No inner loops: sequential DG hits optimal convergence","One-way splitting solves thermo-poroelasticity accurately"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000249,"raw_usage":{"total_tokens":1366,"prompt_tokens":705,"completion_tokens":661,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":449,"completion_tokens_details":{"reasoning_tokens":580}},"tokens_in":449,"tokens_out":661,"duration_ms":6719,"temperature":1.0,"reasoning_tokens":580,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T23:29:35.034663+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a manufactured solution with a Darcy flux strong enough that the exact flux exceeds the cut-off M used in the code. If the scheme keeps optimal order while the condition M(K∇p_h)=K∇p_h is violated on the exact flux, the cut-off requirement is not actually load-bearing; if it loses order or becomes unstable, the unquantified choice of M is the controlling assumption. Compare with the fully implicit DG method of [18] at the same time step to separate splitting error from cut-off error.","supporting_citations":[{"cited_title":"A sequential discontinuous Galerkin method for the cou- pling of flow and geomechanics","cited_arxiv_id":null,"evidence_quote":"Provides the sequential discontinuous Galerkin splitting idea for the Biot model that the paper extends to the three-field thermo-poroelasticity system."},{"cited_title":"Symmetric interior penalty discontinuous Galerkin method for nonlinear fully coupled quasi-static thermo-poroelasticity problems","cited_arxiv_id":null,"evidence_quote":"Supplies the fully implicit SIPDG/backward-Euler scheme used for optimal first-layer values and the consistency lemma underpinning the error equations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the cut-off operator M and splitting solution schemes for the same nonlinear thermo-poroelasticity model; the paper adapts M to a non-iterative sequential scheme."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes well-posedness and the regularity Assumption A for the quasi-static thermo-poroelastic equations, the backdrop for the discrete analysis."},{"cited_title":"F., 2005","cited_arxiv_id":null,"evidence_quote":"Provides the uniform Lipschitz property of the cut-off operator used in existence, uniqueness, and stability estimates."},{"cited_title":"Discontinuous Galerkin methods for solving elliptic and parabolic equa- tions","cited_arxiv_id":null,"evidence_quote":"Supplies the trace and inverse inequalities, elliptic projection estimates, and DG bilinear-form coercivity used throughout the error analysis."},{"cited_title":"N., 1982","cited_arxiv_id":null,"evidence_quote":"Supplies the interior penalty DG coercivity and norm equivalence used for the mechanical and diffusion bilinear forms."},{"cited_title":"DG approximation of coupled Navier-Stokes and Darcy equa- tions by Beaver-Joseph-Saffman interface condition","cited_arxiv_id":null,"evidence_quote":"Supplies the Korn-type inequality used to control the displacement gradient by the DG energy norm."}],"review_version":1}