{"id":"2b631724-f745-4ec4-bb4e-97e6833532b4","arxiv_id":"2412.04231","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A fully discrete finite element and Euler scheme for the 2D no-slip stochastic Navier-Stokes equations converges pathwise in probability at nearly 1.5 spatial and 0.5 temporal order.","lead":"This paper proves that a standard finite element plus Euler scheme for simulating two-dimensional random fluid flow in a box with no-slip walls converges to the true solution at a guaranteed rate. The result, nearly three-halves order in space and half order in time, holds for general multiplicative noise that earlier rate proofs could not handle.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central claim hinges on h-uniform discrete Sobolev embeddings and a discrete log-L∞ bound that are cited rather than fully proved; a hidden h-dependence there would invalidate the temporal rate in Theorem 5.2.","rationale":"The reader's acceptance is reasonable: the proof handles the non-commuting Helmholtz projection via fractional Stokes regularity and stopping-time estimates, and the final probability display balances h and τ carefully. The self-identified limitations (no pressure estimates, no general Lq rates, no 3D extension) are scope restrictions rather than hidden flaws. The single most fragile point is exactly the h-uniform discrete embedding family and the discrete log-L∞ bound, which the reader also identified. I agree with that identification. I did not find an internal inconsistency: the cited results [31] and [49] are standard for Taylor-Hood elements, and the log-L∞ bound is standard in 2D, so the premise is plausible. However, because the proof of Remark 5.1 is compressed and the curved-boundary approximation is not discussed, a concrete verification is worthwhile. Since no failure is demonstrated, the verdict remains ACCEPT and no change from the reader's verdict is needed.","tokens_in":36634,"tokens_out":38809,"duration_ms":378762,"concrete_test":"Compute the three ratios on a family of quasi-uniform meshes for a smooth domain with P3/P2 Taylor-Hood: (i) ||u_h||_{L∞}/||u_h||_{\\dot H^{ρ,2}_h}, (ii) ||∇u_h||_{L^{2/(2-ρ)}}/||u_h||_{\\dot H^{ρ,2}_h}, and (iii) ||u_h||_{L∞}/(sqrt(ln(1/h)) ||∇u_h||_{L2}), using random coefficient vectors normalized so ||u_h||_{\\dot H^{ρ,2}_h}=1, for ρ=1.1, 1.3, 1.49 and h=2^{-3},...,2^{-7}. If the suprema stay bounded in h, the premise is supported; if any grows like h^{-δ}, recompute (5.12)-(5.15) with that δ and check whether the final display in Theorem 5.2 still tends to zero. An analytical re-derivation of (5.18) that tracks every constant from [31] and [49] would be the conclusive version.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 5.2's temporal convergence rests on Theorem 5.1, whose Step 5 uses the discrete log-L∞ inequality ||u_h||_{L∞} ≤ c sqrt(ln(1/h)) ||∇u_h||_{L2}, and whose Steps 2-3 bound I^(2), I^(3), and χ^(3) using the h-independent embeddings from Remark 5.1: ||u_h||_{L∞} ≤ c ||u_h||_{\\dot H^{ρ,2}_h} and ||∇u_h||_{L^{2/(2-ρ)}} ≤ c ||u_h||_{\\dot H^{ρ,2}_h}. If any of these constants actually carries a factor h^{-κ}, then (5.12) and (5.15) gain h^{-κ}, and the last term in Theorem 5.2 becomes roughly h^{-κ} τ^{1-β-γ1}; since τ ≤ h this is at most h^{1-β-γ1-κ}, which diverges as h→0 unless κ is unusually small. The proof of Remark 5.1 is only a sketch: it invokes [31, Theorem 3.1] and [49, Theorem 3.1] together with an inverse estimate, and the log bound is cited to [12, Lemma 4.9.2] without checking the curved-boundary/polygonal-domain setting that the paper leaves implicit. This is the most load-bearing technical premise: it is independent of the convergence architecture and, if false, destroys the claimed h^α + τ^β rate. I do not see a contradiction inside the paper itself, but the premise is not fully demonstrated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the pathwise uniform convergence in probability of a fully discrete finite element method for the two-dimensional stochastic Navier-Stokes equations with multiplicative noise and no-slip boundary conditions. The spatial discretization uses P3/P2 Taylor-Hood elements and the temporal discretization is an implicit Euler scheme. The main result, Theorem 5.2, states that for initial data in L^4_F0(Ω; dot H^{3/2,2}), the maximal-in-time squared L2 error converges to zero in probability at rates arbitrarily close to h^{3/2} in space and τ^{1/2} in time, under the mesh constraint τ≤h. The proof combines new regularity estimates for the SNSE (Proposition 3.1), a spatial semidiscretization error bound (Theorems 4.1-4.3), and a temporal error estimate on a local probability set (Theorem 5.1). The error decomposition is coherent and the final rate follows by balancing the spatial and temporal errors with the probabilities of exceptional events.","tokens_in":36922,"tokens_out":11359,"duration_ms":103811,"significance":"If the advertised rates hold, this is a substantial improvement over the existing literature for no-slip SNSEs: previous finite element analyses either gave no explicit rates or only linear spatial and near-1/4 temporal rates. The main technical novelty is handling the lack of divergence-free diffusion coefficients by working directly with global mild solutions and using maximal Lp-regularity rather than relying on the structure ⟨(u·∇)u, A2u⟩=0. The paper also provides a fairly complete proof of the regularity estimates in Proposition 3.1. The central claims are falsifiable and parameter-free; there are no fitted parameters or circular reductions. The main caveat is that several h-uniform discrete estimates are cited or only sketched, and their proof is the principal point requiring attention before the rates can be considered fully established.","major_comments":[{"comment":"The h-uniform discrete embeddings ||u_h||_{L∞} ≤ c ||u_h||_{dot H^{ρ,2}_h} and ||∇u_h||_{L^{2/(2-ρ)}} ≤ c ||u_h||_{dot H^{ρ,2}_h}, together with the log-L∞ bound ||u_h||_{L∞} ≤ c sqrt(ln(1/h)) ||∇u_h||_{L2}, are load-bearing: they enter directly into the bounds for I^(2), I^(3), and χ^(3) in Theorem 5.1. If any of these constants actually carried a factor h^{-κ}, then (5.12) and (5.15) would gain h^{-κ} and the last term in Theorem 5.2 would become h^{-κ} τ^{1-β-γ1}, which diverges under τ≤h unless κ is unusually small. The proof of Remark 5.1 is only a sketch: it invokes [31, Theorem 3.1] and [49, Theorem 3.1] without addressing the fact that O is a smooth domain while the finite element spaces are constructed on the polygonal union Oh; the cited results are standard for polygonal domains. The log bound is cited to [12, Lemma 4.9.2] without checking its hypotheses in the curved-boundary/polygonal-approximation setting. Please provide a complete proof or precise references that cover the smooth-boundary case, or state the additional geometric assumptions on the mesh that make these estimates h-uniform.","section":"Section 5, Remark 5.1 and Step 5 of Theorem 5.1"},{"comment":"These lemmas contain nontrivial h-uniform estimates used throughout the spatial error analysis, but they are either proved by a one-line spectral argument or reported as \"well-known\". Lemma 4.1(i) and (iii) give h-uniform semigroup and stochastic-convolution bounds in discrete interpolation spaces, and Lemma 4.2(ii) asserts the h-uniform boundedness of the L2 projection Ph in L(dot H^{α,2}, dot H^{α,2}_h) for all α∈[0,2]. The latter is not an immediate consequence of standard estimates for the Stokes projection, especially for fractional α. Since these estimates are essential to Theorems 4.2 and 4.3, please provide a proof or a precise reference for each assertion, or state the extra hypotheses on the mesh and the Taylor-Hood pair needed to make them true.","section":"Section 4, Lemmas 4.1 and 4.2"},{"comment":"The relationship between the smooth domain O and the discrete polygonal domain Oh is not specified. The text says \"conforming and quasi-uniform triangulation of the domain O\" but then defines Oh as the union of elements and extends functions by zero to O\\Oh. For a smooth (or C^{3,1}) boundary, a conforming triangulation with straight elements cannot cover O exactly. The paper should state the boundary approximation used (e.g., that the vertices on ∂Oh lie on ∂O and that the symmetric difference of O and Oh has measure O(h^3) or similar) and explain why the cited results of [13], [31], and [49] remain valid in this setting. Without this, the h-uniform constants in Remark 5.1 and in Lemmas 4.2 and 4.4 are not fully justified.","section":"Section 4, paragraph before Theorem 4.1"}],"minor_comments":[{"comment":"The indicator notation /BD[0,tR,ρ] appears without a subscript in the stochastic convolution term; the notation should be made consistent with the stopping time definition.","section":"Section 2.3, definition of global mild solution"},{"comment":"The final condition is written as \"2/(1+β-α) ≥ q and β<1/q\"; it would be clearer to state it as \"q ≤ 2/(1+β-α) and β<1/q\".","section":"Lemma 3.1"},{"comment":"The formula for R contains nested logarithms and is difficult to parse; please display it more clearly and define all constants consistently.","section":"Proof of Theorem 4.3"},{"comment":"The representation of ψ(t,x) involves a generalized time derivative of a sum of stochastic integrals; the notation is ambiguous and should be clarified.","section":"Remark 5.2"},{"comment":"The roles of the auxiliary sequences ξj,Rh and ηj,Rh in the decomposition Ej,Rh = ξj,Rh + ηj,Rh are not explained intuitively; a short explanatory sentence would improve readability.","section":"Proof of Theorem 5.1, Step 1"},{"comment":"Reference [10] contains a typo in the arXiv number (\"2305.109999\" appears to have six nines); please correct it. Reference [38] is an arXiv preprint; if a published version exists, please cite it.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"This is a strong paper that likely deserves publication after the technical gaps in the discrete embeddings and the domain-approximation setting are addressed. My main reservation is not about the convergence architecture, which is sound, but about the h-uniform estimates that are cited or only sketched. Given that the central claim would collapse if any of these constants carried an h^{-κ} factor, the details should be supplied before acceptance. The paper's fit with the journal is appropriate, and the result would be a significant contribution to the numerical analysis of stochastic Navier-Stokes equations."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper is the first to give explicit near-optimal rates for a fully discrete finite element scheme for the 2D stochastic Navier-Stokes equations with no-slip boundary conditions, without requiring the diffusion coefficients to be divergence-free or vanish on the boundary. That is a genuine advance: prior rate proofs, notably Breit and Prohl, needed those structural assumptions. The spatial rate is nearly h^{3/2} and the temporal rate nearly tau^{1/2}, with convergence in probability pathwise uniformly in time.\n\nWhat the paper does well: the main mathematical difficulty is the non-commuting Helmholtz projection under no-slip boundary conditions. The authors handle it through fractional Sobolev regularity estimates derived from maximal Lp-regularity, and they split the error into deterministic and stochastic convolution parts in a clean way. The stopping-time argument is standard but carefully carried out. They also state the regularity results honestly, including the slow logarithmic-type estimate in Theorem 4.3.\n\nSoft spots, in proportion: several technical lemmas are delegated to references. In particular, Remark 5.1 sketches the h-uniform discrete embeddings and cites the discrete log-L∞ bound to Brenner-Scott without discussing the polygonal approximation of the smooth domain. The stress-test worry is that if these constants carried an h^{-kappa} factor, the temporal rate would degrade. But the derivation in Remark 5.1 is explicit enough that I do not think this is likely to break; it is worth a referee check, not a reason to reject. The abstract omits the tau ≤ h constraint and the H^{3/2} initial-data assumption, which are stated precisely in Theorem 5.2. There are no numerical experiments, but that is not a flaw for a purely theoretical convergence paper. The unconditional rate in Theorem 4.3 is very slow, but it is not load-bearing for the main theorem.\n\nOverall, the central argument holds up. The paper deserves a serious referee; a referee should verify the cited embedding results and the log-L∞ inequality in this setting. I would cite it if I worked on SPDE numerics.","headline":"First explicit convergence rates for fully discrete FE schemes for 2D stochastic Navier-Stokes with no-slip boundary and general multiplicative noise; the proof is credible, with the main caveat being reliance on standard but partly cited discrete Sobolev embeddings.","tokens_in":37481,"tokens_out":2337,"would_cite":true,"duration_ms":24483,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M60","65C30","60H15","60H35","35Q30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a P3/P2 Taylor-Hood plus Euler discretization of the two-dimensional stochastic Navier-Stokes equations with no-slip boundary conditions converges pathwise uniformly in probability, at spatial order arbitrarily…","keywords":["stochastic Navier-Stokes equation","no-slip boundary conditions","P3/P2 Taylor-Hood finite element","Euler scheme","pathwise uniform convergence","convergence in probability","multiplicative noise"],"falsifier":"One concrete way to test the claim is to compute, on a sequence of quasi-uniform P3/P2 Taylor-Hood meshes, the supremum over discrete divergence-free $u_h$ of $\\|u_h\\|_{L^\\infty}/\\|u_h\\|_{\\dot H^{\\rho,2}_h}$; if this ratio is unbounded as $h\\to 0$, the h-independent embedding premise fails. An end-to-end numerical alternative is to manufacture a smooth no-slip solution of (1.1) with multiplicative noise coefficients that do not vanish on $\\partial O$, run the fully discrete scheme with $\\tau \\le h$, and check whether $P\\{\\max_j \\|y(j\\tau)-Y_j\\|_{L^2}^2 \\ge \\varepsilon(h^\\alpha+\\tau^\\beta)\\}$ with $\\alpha=2.9$, $\\beta=0.9$ tends to zero as $h,\\tau \\to 0$; a positive limiting probability would disprove the theorem.","tokens_in":36387,"feed_emoji":"🌊","tokens_out":11707,"duration_ms":103496,"temperature":0.7,"pith_summary":"This paper proves a quantitative error estimate for a fully discrete finite-element method for the two-dimensional stochastic Navier-Stokes equations with multiplicative noise and no-slip boundary conditions. Using P3/P2 Taylor-Hood elements in space and the Euler scheme in time, the numerical solution converges pathwise uniformly in probability to the global mild solution at a rate arbitrarily close to $h^{3/2}$ in space and $\\tau^{1/2}$ in time, under the constraint $\\tau \\le h$. The result matters because earlier finite-element analyses for no-slip stochastic Navier-Stokes either gave no explicit rate or assumed the noise coefficients are divergence-free and vanish at the boundary, whereas the equation studied here keeps the full boundary coupling. The proof splits the error into a spatial semidiscretization part and a temporal discretization part, controlling rare large-deviation events through stopping times and logarithmic-in-probability regularity estimates.","feed_headline":"Stochastic Navier-Stokes: P3/P2 scheme converges at near 3/2 order","feed_subtitle":"No-slip walls plus multiplicative noise: the full scheme also attains near half-order time convergence.","key_machinery":"The argument is carried by the P3/P2 Taylor-Hood finite element pair and its discrete Stokes operator $A_h$, together with a decomposition of the total error into a spatial part $y - y_h$ and a temporal part $y_h(j\\tau)-Y_j$. For the spatial part, stopping times and Gronwall-type estimates convert the mild solution's $\\dot H^{\\rho,2}$ regularity into a nearly-$h^\\rho$ bound with an $\\exp(cR^2)$ constant traded against the rare-event probability $1 - c/\\ln(1+R)$. For the temporal part, the implicit Euler scheme is analyzed through four auxiliary sequences $\\chi^{(1)},\\dots,\\chi^{(4)}$ that isolate the stiffness, the nonlinear increment, the nonlinear mismatch, and the stochastic increment, and their stability is controlled by discrete resolvent estimates and stochastic-convolution maximal inequalities. The h-independent discrete Sobolev embeddings of Remark 5.1---$\\|u_h\\|_{L^\\infty} \\le c \\|u_h\\|_{\\dot H^{\\rho,2}_h}$ and $\\|\\nabla u_h\\|_{L^{2/(2-\\rho)}} \\le c\\|u_h\\|_{\\dot H^{\\rho,2}_h}$---are what keep the temporal error bound free of negative powers of $h$.","core_discovery":"The central claim is Theorem 5.2: for initial data $y_0 \\in L^4_{\\mathcal F_0}(\\Omega; \\dot H^{3/2,2})$, any $\\alpha \\in (2,3)$, $\\beta \\in (0,1)$, and any $\\varepsilon > 0$, the probability that $\\max_{1\\le j\\le J} \\|y(j\\tau)-Y_j\\|_{L^2}^2$ exceeds $\\varepsilon (h^\\alpha + \\tau^\\beta)$ tends to zero as $h \\to 0$, $\\tau \\to 0$, with $\\tau \\le h$. In words, the full discretization converges pathwise uniformly in probability with spatial order arbitrarily close to $3/2$ and temporal order arbitrarily close to $1/2$. These rates are shown for the genuine no-slip problem, where the Helmholtz projection does not preserve the zero-trace condition and the standard cancellation identity used under periodic boundary conditions fails. The paper also establishes the underlying regularity of the global mild solution in $\\dot H^{\\rho,2}$, $\\rho \\in (1,3/2)$, with tail probability $1 - c/\\ln(1+R)$.","pith_inferences":["The rates saturate at the solution's regularity rather than at the polynomial degree: if the mild solution were known to live in $\\dot H^{3/2+\\delta,2}$ with high probability, the same argument would push the spatial rate above $3/2$ toward the cubic element's formal order.","The same regime-splitting machinery---trading an $\\exp(cR^2)$ error constant against a $1/\\ln(1+R)$ probability---should transfer to other stochastic PDEs whose solutions have only logarithmic tail regularity, including variants with transport noise or gradient-dependent coefficients.","One direct numerical check would be to manufacture a no-slip solution whose noise coefficients do not vanish on the boundary and measure the empirical rate of $\\max_j \\|y(j\\tau)-Y_j\\|_{L^2}$ under $\\tau \\le h$; agreement with exponents approaching $3/2$ and $1/2$ would confirm that the h-independent discrete embeddings hold with reasonable constants on practical meshes."],"forward_implications":["For every $\\alpha < 3$ and $\\beta < 1$, the probability that the pathwise-uniform $L^2$ error exceeds $\\varepsilon(h^\\alpha+\\tau^\\beta)$ goes to zero as $h,\\tau\\to 0$ with $\\tau \\le h$, so the method is certified at nearly optimal rates for no-slip stochastic Navier-Stokes.","The semidiscrete estimate $\\|y - y_h\\|_{L^p(\\{t_{R,\\rho}=T\\}; C([0,T];L^2))} \\le c h^\\rho \\ln(1/h) \\exp(cR^2)$ gives a quantitative trade-off between spatial accuracy and the size of the probability set on which it holds.","The temporal estimate (5.3) bounds the mean squared pathwise error between the semidiscrete and fully discrete solutions by $c\\tau \\exp(cR_h^2)(1+\\ln(1/h)R_{h,\\tau}^2)$, yielding nearly order-$1/2$ temporal convergence after optimizing the free parameters.","As noted in Remark 5.2, the auxiliary pressure $\\phi$ inherits pathwise uniform convergence in probability, while the physical pressure $\\psi$ does not because of its limited temporal regularity."],"supporting_citations":[{"why":"Establishes the previous best convergence rates in probability for fully discrete no-slip SNSE under divergence-free boundary-vanishing noise; the paper's target is to remove that structural assumption.","marker":"[11]"},{"why":"Describes the stopping-time and regular/irregular splitting methodology for nearly $1/2$-order temporal convergence that the full-discretization analysis adapts.","marker":"[9]"},{"why":"Provides the finite-element discretization framework for multiplicative noise and the well-posedness/stability of the implicit Euler scheme used here.","marker":"[15]"},{"why":"Supplies the quasi-local interpolation operator preserving discrete divergence, used to prove the h-independent discrete Sobolev embeddings in Remark 5.1.","marker":"[31]"},{"why":"Defines the P3/P2 Taylor-Hood spaces and their Stokes stability, the spatial method whose optimal approximation properties the error analysis uses.","marker":"[49]"},{"why":"Gives the stability theorem for higher-order Hood-Taylor methods used to bound the discrete Stokes projection error in Lemma 4.4.","marker":"[13]"},{"why":"Provides the maximal $L^p$-regularity theory for stochastic evolution equations that yields the mild solution's $\\dot H^{\\rho,2}$ regularity and tail estimates.","marker":"[54]"},{"why":"Establishes global mild-solution existence for the stochastic Navier-Stokes problem in UMD Banach spaces, the solution concept on which all convergence statements are based.","marker":"[53]"},{"why":"Supplies the discrete maximal regularity for implicit parabolic difference equations used in Lemma 5.1 to stabilize the temporal error sequences.","marker":"[1]"},{"why":"Provides maximal inequalities for stochastic convolutions and pathwise uniform error estimates for time discretizations, used in Lemma 5.2 for the noise increment.","marker":"[55]"}],"fun_headline_variants":["No-slip stochastic Navier-Stokes: near-3/2 order proven","2D stochastic Navier-Stokes: pathwise uniform near-3/2 convergence","Stochastic Navier-Stokes no-slip: finite element order near 3/2","Near-3/2 spatial rate for stochastic Navier-Stokes with no-slip","Stochastic Navier-Stokes no-slip: near 3/2 space, 1/2 time"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The rate proof requires that the discrete Sobolev embedding inequalities on the Taylor-Hood spaces hold with constants independent of mesh size h, including the logarithmic L-infinity bound used in Step 5; if these constants grow as h shrinks, the temporal error bound acquires negative powers of h and Theorem 5.2's joint limit would not follow from this argument.","fun_headline_variants_meta":{"raw":{"variants":["No-slip stochastic Navier-Stokes: near-3/2 order proven","2D stochastic Navier-Stokes: pathwise uniform near-3/2 convergence","Stochastic Navier-Stokes no-slip: finite element order near 3/2","Near-3/2 spatial rate for stochastic Navier-Stokes with no-slip","Stochastic Navier-Stokes no-slip: near 3/2 space, 1/2 time"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001295,"raw_usage":{"total_tokens":5236,"prompt_tokens":846,"completion_tokens":4390,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":462,"completion_tokens_details":{"reasoning_tokens":4274}},"tokens_in":462,"tokens_out":4390,"duration_ms":32297,"temperature":1.0,"reasoning_tokens":4274,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:38:22.123271+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete way to test the claim is to compute, on a sequence of quasi-uniform P3/P2 Taylor-Hood meshes, the supremum over discrete divergence-free $u_h$ of $\\|u_h\\|_{L^\\infty}/\\|u_h\\|_{\\dot H^{\\rho,2}_h}$; if this ratio is unbounded as $h\\to 0$, the h-independent embedding premise fails. An end-to-end numerical alternative is to manufacture a smooth no-slip solution of (1.1) with multiplicative noise coefficients that do not vanish on $\\partial O$, run the fully discrete scheme with $\\tau \\le h$, and check whether $P\\{\\max_j \\|y(j\\tau)-Y_j\\|_{L^2}^2 \\ge \\varepsilon(h^\\alpha+\\tau^\\beta)\\}$ with $\\alpha=2.9$, $\\beta=0.9$ tends to zero as $h,\\tau \\to 0$; a positive limiting probability would disprove the theorem.","supporting_citations":[{"cited_title":"Breit and A","cited_arxiv_id":null,"evidence_quote":"Establishes the previous best convergence rates in probability for fully discrete no-slip SNSE under divergence-free boundary-vanishing noise; the paper's target is to remove that structural assumption."},{"cited_title":"Breit and A","cited_arxiv_id":null,"evidence_quote":"Describes the stopping-time and regular/irregular splitting methodology for nearly $1/2$-order temporal convergence that the full-discretization analysis adapts."},{"cited_title":"Brze´ zniak, E","cited_arxiv_id":null,"evidence_quote":"Provides the finite-element discretization framework for multiplicative noise and the well-posedness/stability of the implicit Euler scheme used here."},{"cited_title":"Girault and L","cited_arxiv_id":null,"evidence_quote":"Supplies the quasi-local interpolation operator preserving discrete divergence, used to prove the h-independent discrete Sobolev embeddings in Remark 5.1."},{"cited_title":"Stenberg","cited_arxiv_id":null,"evidence_quote":"Defines the P3/P2 Taylor-Hood spaces and their Stokes stability, the spatial method whose optimal approximation properties the error analysis uses."},{"cited_title":"Brezzi and R","cited_arxiv_id":null,"evidence_quote":"Gives the stability theorem for higher-order Hood-Taylor methods used to bound the discrete Stokes projection error in Lemma 4.4."},{"cited_title":"van Neerven, M","cited_arxiv_id":null,"evidence_quote":"Provides the maximal $L^p$-regularity theory for stochastic evolution equations that yields the mild solution's $\\dot H^{\\rho,2}$ regularity and tail estimates."},{"cited_title":"van Neerven, M","cited_arxiv_id":null,"evidence_quote":"Establishes global mild-solution existence for the stochastic Navier-Stokes problem in UMD Banach spaces, the solution concept on which all convergence statements are based."},{"cited_title":"Ashyralyev and P","cited_arxiv_id":null,"evidence_quote":"Supplies the discrete maximal regularity for implicit parabolic difference equations used in Lemma 5.1 to stabilize the temporal error sequences."},{"cited_title":"van Neerven and M","cited_arxiv_id":null,"evidence_quote":"Provides maximal inequalities for stochastic convolutions and pathwise uniform error estimates for time discretizations, used in Lemma 5.2 for the noise increment."}],"review_version":1}