{"id":"11e1810c-7fbf-4eae-bc92-b8c88a598f62","arxiv_id":"2606.07164","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"A regularised fully discrete finite element scheme for the stochastic Cahn-Hilliard equation with singular potential converges to the pathwise unique strong solution as discretization parameters vanish.","lead":"The paper develops a regularised finite element scheme for numerically solving the stochastic Cahn-Hilliard equation with a singular potential and shows its convergence to the true solution. Smart generalists might read it to see how numerical methods can reliably handle singular nonlinearities and noise in models of phase separation.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Uniform H¹-estimate for the regularised problem must be shown independent of the regularisation parameter ε to justify the final singular limit.","rationale":"The reader’s weakest assumption is exactly the step that closes the argument; the full text does not remove the need to verify uniformity of that H¹ bound with respect to the regularisation parameter. Because the manuscript is a convergence proof, the verdict moves from UNVERDICTED to CONDITIONAL pending confirmation that the constant is indeed independent of ε.","tokens_in":1638,"tokens_out":386,"duration_ms":13245,"concrete_test":"Locate the section proving the uniform H¹ estimate for the regularised problem. Extract the constant C_ε appearing in the bound ||u_ε||_{L^∞(0,T;H¹)} ≤ C_ε and trace its dependence on the regularisation parameter through the energy estimate and the application of Itô’s formula. If C_ε remains bounded as ε→0 the argument is intact; if it diverges the passage to the singular limit requires additional justification.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim proceeds in two steps: (i) monotonicity arguments yield convergence of the fully discrete scheme to a regularised (still singular-potential) equation for fixed regularisation parameter, then (ii) a uniform-in-ε H¹ bound on the regularised solutions is invoked to pass to the original singular equation. The second step is load-bearing because any ε-dependence hidden in the constant of the H¹ estimate would prevent extraction of a limit that satisfies the double-obstacle constraint in the required sense. In the stochastic multiplicative-noise setting this bound is typically obtained via Itô calculus on a regularised energy; the noise term and the singular-potential approximation can each introduce ε-dependent constants that are not obviously controlled uniformly.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript develops a regularised fully discrete finite element scheme for the stochastic Cahn-Hilliard equation with singular double-obstacle potential and multiplicative conservative noise. It establishes stability estimates uniform in the discretization parameters, proves convergence of the scheme to a regularised (still singular-potential) equation via monotonicity arguments, and then invokes a uniform H¹ estimate on the regularised solutions to pass to the limit as the regularization parameter vanishes, obtaining convergence to the pathwise-unique probabilistically strong solution of the original equation. Numerical simulations comparing the regularised and unregularised approximations are included.","tokens_in":1816,"tokens_out":471,"duration_ms":12743,"significance":"If the uniform H¹ estimate is established independently of the regularization parameter, the result supplies a rigorous convergence theory for numerical approximation of a singular stochastic PDE that is otherwise difficult to treat directly; the combination of monotonicity-based convergence for the regularised problem with a uniform energy bound is a standard but technically demanding route in this area and would be a useful addition to the literature on numerical SPDEs.","major_comments":[{"comment":"The passage from the regularised to the original singular equation rests on a uniform H¹ estimate for the regularised problem that is independent of the regularization parameter ε (invoked after the monotonicity step). In the multiplicative-noise setting this bound is typically derived via Itô calculus applied to a regularised energy; the proof must explicitly control any ε-dependent contributions arising from the noise term and from the approximation of the singular potential, otherwise the limit may fail to satisfy the double-obstacle constraint in the required sense.","section":"uniform H¹-estimate (abstract and the section establishing the singular limit)"}],"minor_comments":[{"comment":"The precise form of the regularization of the singular potential (e.g., the specific mollification or penalty function) should be stated explicitly in the abstract and in the statement of the regularised problem.","section":null},{"comment":"Notation for the fully discrete scheme (time-step, spatial mesh size, regularization parameter) should be introduced once and used consistently throughout the stability and convergence statements.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and constructive feedback. We address the single major comment below.","responses":[{"response":"We appreciate this observation. The uniform H¹ estimate is derived in Section 4 by applying Itô's formula to the regularised energy. The multiplicative conservative noise term produces no ε-dependent blow-up because the noise is divergence-free and the regularisation preserves the zero-mean constraint; the resulting stochastic integral is controlled via Burkholder-Davis-Gundy and the uniform L² bound already obtained from the monotonicity argument. The approximation of the singular potential contributes only non-positive terms that vanish as ε→0 by construction of the regularisation. Consequently the double-obstacle constraint is recovered in the limit. We will add an explicit remark after the statement of the uniform estimate clarifying these controls.","revision_made":"yes","referee_comment":"[uniform H¹-estimate (abstract and the section establishing the singular limit)] The passage from the regularised to the original singular equation rests on a uniform H¹ estimate for the regularised problem that is independent of the regularization parameter ε (invoked after the monotonicity step). In the multiplicative-noise setting this bound is typically derived via Itô calculus applied to a regularised energy; the proof must explicitly control any ε-dependent contributions arising from the noise term and from the approximation of the singular potential, otherwise the limit may fail to satisfy the double-obstacle constraint in the required sense."}],"tokens_in":1294,"tokens_out":317,"duration_ms":16587,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper gives a fully discrete finite-element scheme for the stochastic Cahn-Hilliard equation with singular double-obstacle potential and conservative multiplicative noise. It regularizes the potential, proves uniform stability, uses monotonicity to pass to the regularized equation, and invokes a uniform H1 bound to reach the original singular problem.\n\nThis is a direct extension of earlier deterministic or non-singular schemes to the stochastic singular case. The combination of regularization, monotonicity arguments, and the specific noise structure is new content.\n\nThe stability estimates uniform in discretization parameters are standard and credible for this class of problems. The convergence claim to the pathwise-unique strong solution follows the usual pattern once the pieces are in place.\n\nThe load-bearing step is the uniform-in-ε H1 estimate on the regularized solutions. With multiplicative noise, Itô calculus on the energy typically produces terms whose constants can depend on the regularization; if those are not controlled independently of ε, the singular limit does not go through. The abstract states the bound exists but does not indicate how the noise contribution is handled uniformly.\n\nThe work is aimed at researchers in numerical SPDEs who already work on phase-field models. It is a solid technical piece that deserves a serious referee, provided the H1 estimate is checked in detail. I would send it to review.","headline":"A convergent regularized finite-element scheme for the singular stochastic Cahn-Hilliard equation, with the uniform H1 bound as the key unverified step.","tokens_in":2310,"tokens_out":346,"would_cite":false,"duration_ms":10207,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A regularized finite element scheme converges to the pathwise unique strong solution of the singular stochastic Cahn-Hilliard equation.","keywords":["stochastic Cahn-Hilliard equation","singular potential","finite element approximation","numerical convergence","regularization","multiplicative noise","monotonicity method"],"falsifier":"Numerical evidence that the H^1 norm of solutions to the regularised problem grows unbounded as the regularization parameter tends to zero would disprove the passage to the singular limit.","tokens_in":2550,"feed_emoji":"📊","tokens_out":534,"duration_ms":14597,"temperature":0.7,"pith_summary":"The paper constructs a regularized fully discrete finite element scheme for the stochastic Cahn-Hilliard equation that includes a singular double-obstacle potential and multiplicative conservative noise. Stability estimates hold uniformly in the discretization parameters, allowing convergence first to a regularized equation via monotonicity arguments and then to the original singular equation via a uniform H^1 bound. This matters for obtaining reliable approximations when the singular potential prevents direct numerical treatment of the limiting problem.","feed_headline":"Regularized scheme converges for singular stochastic Cahn-Hilliard","feed_subtitle":"Uniform H^1 bound lets the finite element approximation reach the unique strong solution of the singular problem.","key_machinery":"The regularised fully discrete finite element approximation scheme, whose uniform H^1-estimate independent of regularization and discretization parameters enables passage from the monotonicity limit to the singular limit.","core_discovery":"Thanks to a uniform H^1-estimate for the regularised problem the regularised solution converges towards the pathwise unique probabilistically strong solution of the original singular stochastic Cahn--Hilliard equation.","pith_inferences":["The uniform stability technique could be tested on related singular stochastic phase-field models with different noise structures.","If the H^1 bound persists under weaker noise assumptions, the method might extend to equations with non-conservative multiplicative noise."],"forward_implications":["The scheme produces stable approximations that remain controlled as mesh size and time step vanish.","Convergence holds pathwise to the unique probabilistically strong solution.","Numerical experiments can directly compare the regularised scheme against its unregularised counterpart and display the influence of the conservative noise."],"fun_headline_variants":["Regularized FEM converges to singular stochastic Cahn-Hilliard solution","Convergence proven for regularized singular stochastic Cahn-Hilliard equation","Finite element approximation converges to unique stochastic Cahn-Hilliard solution","Regularization allows convergence to singular potential Cahn-Hilliard solution"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The regularised problem admits a uniform H^1-estimate independent of the regularization and discretization parameters.","fun_headline_variants_meta":{"raw":{"variants":["Regularized FEM converges to singular stochastic Cahn-Hilliard solution","Convergence proven for regularized singular stochastic Cahn-Hilliard equation","Finite element approximation converges to unique stochastic Cahn-Hilliard solution","Regularization allows convergence to singular potential Cahn-Hilliard solution"]},"model":"grok-4.3","cost_usd":0.008522,"raw_usage":{"total_tokens":3794,"prompt_tokens":555,"num_sources_used":0,"completion_tokens":74,"cost_in_usd_ticks":85224500,"prompt_tokens_details":{"text_tokens":555,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3165,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":555,"tokens_out":74,"duration_ms":15835,"temperature":1.0,"reasoning_tokens":3165,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T21:16:19.194530+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Numerical evidence that the H^1 norm of solutions to the regularised problem grows unbounded as the regularization parameter tends to zero would disprove the passage to the singular limit.","supporting_citations":[],"review_version":1}