{"id":"d5c486b1-aa73-43dd-8d1f-18fcdc75bd62","arxiv_id":"2411.17903","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"An adaptive two-grid preconditioner with fixed-operator implicit-explicit time stepping gives contrast-robust iteration counts for mixed-dimensional shale gas transport in fractured media.","lead":"This paper builds a numerical solver for shale gas flow through fractured rock, combining a fixed linear implicit time step with an adaptive spectral coarse-grid preconditioner. It reports fast iterative convergence that stays robust across fracture-matrix permeability contrasts of 10^3 to 10^9.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Stability theorem's hypotheses do not match the constructed splitting: S(nl) and D(nl) are nonpositive, while the required discrete matrix inequalities are never verified, so the unconditional-stability and fixed-operator O(1)-iteration claims hinge on an unproven concentration-range assumption.","rationale":"I agree with the reader that the weakest assumption is the unverified dominance condition S(lin)>S(nl) and D(lin)>D(nl), together with the mismatch between Theorem 1's hypotheses and the constructed splitting. This is the most load-bearing issue because the entire strategy of building one fixed offline preconditioner and reusing it for every time step requires the fixed linear operator to dominate the nonlinear residual at all times. The two-grid convergence scaling in (19) is also under-proved, but it is secondary: the numerical experiments directly demonstrate practical contrast-robust behavior, whereas a failure of the stability/dominance condition would undermine the method itself. The reader's CONDITIONAL verdict remains appropriate: the concern is real and would be settled by the proposed numerical verification, but it is not evidence of a false result. I would not change the verdict.","tokens_in":23700,"tokens_out":8665,"duration_ms":85558,"concrete_test":"Re-run Test 1a and Test 2b with the authors' Python/Scipy setup, and at every time step after solving (10): (i) compute min and max of all c_m^n and c_f^n and compare them with [cmin,cmax]; (ii) assemble Slin-S(c^n) and Dlin-D(c^n) and compute the smallest eigenvalue, or attempt a sparse Cholesky factorization, of the full mixed-dimensional matrices; (iii) repeat with a stronger source/sink term that drives concentrations below cmin or above cmax. If any concentration leaves [cmin,cmax], or any of these difference matrices is not positive semidefinite, the dominance conditions behind Theorem 1 and the fixed-preconditioner construction are violated. If all checks pass, the gap is a correctable theorem-statement mismatch and the numerical claims stand for those setups.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing point is the stability/dominance condition behind the fixed-operator scheme, not the two-grid theory. In Section 3.2, the authors choose S(lin) and D(lin) from pointwise coefficient upper bounds a*_m=am(x,cmin), b*_m=bm1(x,cmin)+bm2(x,cmax), and b*_f=bf(x,cmax), so S(c^n)<=S(lin) and D(c^n)<=D(lin) in the Loewner order. Consequently S(nl)=S(c^n)-S(lin)<=0 and D(nl)=D(c^n)-D(lin)<=0, whereas Theorem 1 states its hypotheses for symmetric positive definite S(nl),D(nl)>0. Thus Theorem 1 as stated cannot be applied to the constructed scheme; the proof's final condition S(lin)-S(nl)>0 and D(lin)-D(nl)>0 is what is actually needed. That condition is only guaranteed if every nodal concentration remains in [cmin,cmax] for all n, an assumption stated in Section 3.2 but never checked or derived. If a concentration leaves the interval, which is not ruled out by the fracture source/sink terms or by the nonlinear coefficients, the fixed matrix A=S(lin)+tau*D(lin) may fail to dominate the nonlinear residual. Then the unconditional stability result, the offline construction of the spectral coarse space from this fixed A, and hence the headline O(1)-iteration/one-preconditioner claim lose their proven basis. The numerical tables show the scheme works for the tested cases, but they do not verify the discrete matrix inequalities (11) on the full mixed-dimensional system.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a linearly implicit (ImEx) time integration scheme for a mixed-dimensional shale gas transport model with explicit fracture networks, combined with an adaptive spectral two-grid preconditioner for the resulting fixed-matrix linear systems. The linear operator is built once from pointwise coefficient upper bounds, so multiscale basis functions are computed offline only once, and each time step reduces to solving a linear system with a previously constructed preconditioner. The paper claims unconditional stability of the time scheme under conditions on the operator splitting, and claims iterative convergence independent of the fracture/matrix permeability contrast. Numerical experiments cover 30 and 160 fracture networks with homogeneous and heterogeneous matrices over three permeability contrasts.","tokens_in":24097,"tokens_out":6197,"duration_ms":54751,"significance":"If the stability and convergence claims hold, the fixed-operator, one-time-preconditioner approach would be a practically valuable advance: it avoids rebuilding multiscale basis functions every nonlinear iteration or time step, which is the dominant cost in GMsFEM-type solvers for time-dependent problems. The numerical study is extensive, with two fracture geometries, homogeneous and heterogeneous backgrounds, three permeability contrasts, two coarse grids, and both fixed and adaptive numbers of spectral basis functions; the observed iteration counts are low and largely robust across these variations. The adaptive eigenvalue threshold gives a single-parameter mechanism for choosing local coarse-space dimension, which is an attractive practical feature. However, as detailed below, the stability theorem as stated does not apply to the constructed splitting, and the concentration-range condition on which the argument relies is never verified; the theoretical two-grid bound also does not itself establish contrast independence. These gaps affect the two central claims of the paper, so substantial revision is needed.","major_comments":[{"comment":"Theorem 1 states hypotheses for symmetric positive definite S(nl)(c^n) and D(nl)(c^n) and gives condition (11) as sufficient for unconditional stability. The constructed splitting, however, uses coefficient-wise upper bounds a*_m = a_m(x,cmin), b*_m = b_m1(x,cmin)+b_m2(x,cmax), and b*_f = b_f(x,cmax), which implies S(c^n) <= S(lin) and D(c^n) <= D(lin) in the Loewner order. Consequently S(nl) = S(c^n) - S(lin) and D(nl) = D(c^n) - D(lin) are negative semidefinite, not positive definite. The theorem as stated is therefore inapplicable to the scheme actually analyzed and tested. The appendix's proof derives the correct sufficient condition, S(lin)-S(nl)(c^n) > 0 and D(lin)-D(nl)(c^n) > 0, which is weaker and is what should appear in the theorem statement. This is a load-bearing mismatch because the abstract and conclusion assert unconditional stability under the stated splitting.","section":"Section 3.2, Theorem 1, and Appendix"},{"comment":"The stability condition ultimately rests on the assumption that all nodal concentrations satisfy cmin <= c_m^n, c_f^n <= cmax for every time step n. The paper states this assumption but never proves a discrete maximum principle, nor does it verify the resulting discrete matrix inequalities (11) for the computed solutions in the numerical tests. The fracture source/sink term f_f(c_f) = c_w ZRT κ_w (c_w - c_f)/mu is nonlinear and can act as a source when c_f < c_w, and the explicit treatment of the nonlinear residual provides no mechanism to enforce the bound. Since the invariance of the concentration range underlies both the unconditional stability claim and the offline construction of the fixed preconditioner A = S(lin) + tau D(lin), the authors should either prove that the scheme preserves the interval [cmin, cmax] or add a direct numerical verification for all reported test cases, and if necessary introduce a projection or clipping step that preserves the stability argument.","section":"Section 3.2 and Section 5"},{"comment":"The two-grid convergence bound K_TG = c_m H^2 / Lambda* is asserted for the matrix A in (13), which is the fixed time-stepping matrix S(lin)+tau D(lin) of the mixed-dimensional system. The derivation combines the local projection estimate (16) with a partition-of-unity argument, but it does not establish that the local eigenproblems (14) with a 'modified diagonal part' are consistent with the global matrix A, nor does it quantify how Lambda* and the constant c_m depend on the fracture permeability contrast and on the time step tau. As written, inequality (19) does not prove contrast-independent convergence; such independence is only an empirical observation in Tables 3 and 4. The authors should either make the dependence on contrast explicit in the bound or present the theoretical result as conditional on the spectral coarse space capturing the relevant fracture-contrast modes.","section":"Section 4.3, Eq. (19)"},{"comment":"The claim of 'iterative convergence independent of the contrast of fracture and porous matrix permeability' is stronger than the data and theory support. For instance, Table 4, Test 2a on the 20x20 coarse grid with m=4 basis functions shows average iterations of 12.5, 28.0, and 72.0 for contrasts 10^3, 10^6, and 10^9, respectively, and the adaptive rows also show a noticeable (albeit milder) increase with contrast (e.g., 14.5, 16.7, 21.0 for delta_lambda=10^-4). The authors should qualify the claim, for example as 'robust' or 'weakly dependent on contrast' in the adaptive regime, or state precisely which configurations demonstrate independence.","section":"Abstract and Conclusion, Tables 3-4"}],"minor_comments":[{"comment":"In the parameters paragraph, 'kf = 10^3, 10^6 and 10^6' appears to be a typo; the subsequent text and tables indicate the intended contrasts are 10^3, 10^6, and 10^9.","section":"Section 5"},{"comment":"The symbol S_n is used in the sentence 'for X n = Bn and X n = Sn' but is never defined; it should presumably be S(lin)+S(nl)(c^n) or the corresponding mass-type matrix from (10).","section":"Appendix"},{"comment":"There are several typographical errors, for example 'particulaly' (Section 3.2), 'porious media' (Section 4), and 'we proof Theorem 1' (Appendix); these should be corrected in revision.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the journal's scope in numerical analysis and scientific computing, and the practical idea of fixing the linear operator to enable one-time construction of a spectral coarse space is attractive. The main obstacle is the mismatch between Theorem 1's hypotheses and the constructed splitting, plus the unverified concentration-range assumption; both are fixable in a revision. The two-grid theory also needs to be tightened so that the stated bound honestly reflects the empirical contrast robustness. I do not see grounds for rejection, but the revision should be substantive rather than cosmetic."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take on arXiv:2411.17903. It's a solid, useful integration of two known building blocks: the fixed-operator linearly implicit (EIN-style) time scheme and an adaptive spectral two-grid preconditioner, applied to a mixed-dimensional shale gas model. The numerical study is reasonably thorough—two fracture networks, three permeability contrasts, two coarse grids, both fixed and adaptive basis counts—and the adaptive thresholding consistently gives O(1) PCG iterations. That part is genuine and worth taking seriously.\n\nThe main new idea is that because the implicit operator is frozen for all time, the spectral coarse space (built from local eigenvectors of that same operator) can be constructed once, offline. That's a nice practical point, and the stability proof in the appendix is a real attempt to justify the splitting. Credit where due: the paper ships a proof, not just numerics.\n\nNow the soft spots, in rough order of importance. The theorem statement does not match the construction. Theorem 1 requires S(nl) and D(nl) to be positive definite, but the splitting built from pointwise upper bounds makes them nonpositive (S(c) ≤ S(lin), D(c) ≤ D(lin) in Loewner order). So the theorem as stated cannot be applied to the scheme. The appendix's actual working condition is S(lin)−S(nl)>0 and D(lin)−D(nl)>0, which is automatically satisfied for the constructed splitting. This is a presentation error, easily fixed by restating the theorem, not a hidden counterexample. Still, it needs fixing.\n\nThe bigger gap is the concentration-range assumption. The upper bounds a*_m, b*_m, b*_f are chosen under the assumption that all nodal concentrations stay in [cmin, cmax]. The paper states this but never verifies it numerically or proves an invariant. If a concentration leaves the interval, the fixed operator may fail to dominate the nonlinear residual, and the unconditional-stability claim loses its foundation. The experiments don't show blow-up, but the claim is stronger than the evidence.\n\nThe two-grid convergence estimate also contains a scaling step that is asserted rather than shown: the eigenvalues are said to scale with H^{-2}, giving ||v−Pv||^2_D ≤ H^2/Λ* ||v||^2_A. That's standard in the GMsFEM literature, but here it's a one-liner. For a paper that states a convergence estimate as a contribution, this deserves more detail.\n\nFinally, the abstract's 'independent of the contrast' is too strong. The tables show iteration counts creeping up with contrast—e.g., Test 2a with m=12 goes 8.3, 10.4, 14.1. That's still O(1) and robust, but it's not flat. Also, there are no timing comparisons or code release, so the efficiency claim rests on iteration counts alone.\n\nOverall: this is a useful paper for researchers working on multiscale preconditioners for nonlinear transport in fractured media. The core approach is sound, the flaws are addressable, and it deserves a serious referee. I'd send it to review, with a request to fix the theorem statement, address the concentration-range assumption, and soften the contrast-independence claim.","headline":"Useful integration of fixed-operator ImEx with adaptive spectral two-grid; the stability proof has a fixable hypothesis mismatch and an unverified concentration-range assumption.","tokens_in":24596,"tokens_out":4426,"would_cite":true,"duration_ms":38566,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M60","65F08","65N55","76S05","65M12"],"pacs":[],"model":"deepseek-v4-flash","headline":"Shale gas transport in fractured media can be advanced in time with a fixed linear operator and solved with an adaptive spectral two-grid preconditioner whose iteration count is independent of fracture-matrix permeability contrast.","keywords":["shale gas transport","fractured porous media","mixed-dimensional model","linearly implicit scheme","explicit-implicit-null method","adaptive spectral coarse space","two-grid preconditioner","high-contrast permeability"],"falsifier":"At each time step of the reported 30- and 160-fracture runs, compute the smallest eigenvalue of $S^{(\\mathrm{lin})}-S^{(\\mathrm{nl})}(c^n)$ and of $D^{(\\mathrm{lin})}-D^{(\\mathrm{nl})}(c^n)$; if any eigenvalue is negative while $c^n$ lies inside $[c_{\\min}, c_{\\max}]$, the unconditional-stability claim fails. Separately, fix the threshold $\\delta_\\lambda=10^{-3}$ and increase the fracture-matrix permeability contrast beyond $10^{9}$; if the average number of PCG iterations rises systematically with contrast, the contrast-independence claim fails.","tokens_in":23517,"feed_emoji":"⛽","tokens_out":13056,"duration_ms":108803,"temperature":0.7,"pith_summary":"Shale gas moves through a porous matrix and a network of thin fractures with very different permeabilities, which makes simulation stiff and expensive. The paper tries to establish two claims: that the nonlinear transport equations can be advanced in time with a fixed linear operator chosen once, giving an unconditionally stable linearly implicit scheme under a dominance condition; and that the resulting linear system can be solved by an adaptive spectral two-grid preconditioner whose iteration count does not grow with fracture-matrix permeability contrast. If both claims hold, each time step reduces to one preconditioned conjugate gradient solve with a preconditioner built once offline, instead of rebuilding a multiscale solver whenever the concentration field changes. The paper supports the claims with tests on 30 and 160 fracture networks, contrasts from $10^{3}$ to $10^{9}$, and both homogeneous and heterogeneous matrix properties.","feed_headline":"One fixed operator tames high-contrast shale gas flow","feed_subtitle":"Fixed-operator time stepping plus an adaptive spectral coarse space keeps PCG solves at O(1) iterations up to contrast 10^9.","key_machinery":"The load-bearing mechanism is the combination of two ingredients. First, the additive split $S(c)=S^{(\\mathrm{lin})}+S^{(\\mathrm{nl})}(c)$, $D(c)=D^{(\\mathrm{lin})}+D^{(\\mathrm{nl})}(c)$, with $S^{(\\mathrm{lin})}, D^{(\\mathrm{lin})}$ chosen as coefficient-wise upper bounds over $[c_{\\min},c_{\\max}]$, converts the nonlinear evolution into one linear system $A=S^{(\\mathrm{lin})}+\\tau D^{(\\mathrm{lin})}$ per time step; Theorem 1 and the appendix's canonical three-step representation give the unconditional-stability criterion $S^{(\\mathrm{lin})}>S^{(\\mathrm{nl})}$, $D^{(\\mathrm{lin})}>D^{(\\mathrm{nl})}$. Second, the two-grid preconditioner uses a symmetric Gauss-Seidel smoother, spectrally equivalent to the diagonal of $A$, and a spectral coarse space generated by the local generalized eigenvalue problem (14), with adaptive thresholding $\\lambda_{m_i}<\\delta_\\lambda$; partition-of-unity multiplication gives continuous basis functions, and the local approximation bound (16) feeds the global estimate $\\|v-Pv\\|_D^2 \\le (H^2/\\Lambda^*)\\|v\\|_A^2$ that yields $K_{TG}=c_m H^2/\\Lambda^*$.","core_discovery":"On its own terms, the central discovery is that an additive split of the operators, $S(c)=S^{(\\mathrm{lin})}+S^{(\\mathrm{nl})}(c)$ and $D(c)=D^{(\\mathrm{lin})}+D^{(\\mathrm{nl})}(c)$, can be tuned so the linear part is fixed for all time steps and yet dominates the nonlinear part. The authors choose $a^*_m(x)=a_m(x,c_{\\min})$, $b^*_m(x)=b_{m1}(x,c_{\\min})+b_{m2}(x,c_{\\max})$, and $b^*_f(x)=b_f(x,c_{\\max})$, then prove in Theorem 1 that the linearly implicit scheme (10) is unconditionally stable whenever $S^{(\\mathrm{lin})}>S^{(\\mathrm{nl})}(c^n)$ and $D^{(\\mathrm{lin})}>D^{(\\mathrm{nl})}(c^n)$ at every step; the appendix verifies positivity of $R_n - \\tfrac{1}{4}D_n$ through a canonical three-step energy estimate. For the linear solve, the coarse space is built from the local generalized eigenvalue problem $A^{\\omega_i}\\psi^{\\omega_i}_k = \\lambda^{\\omega_i}_k D^{\\omega_i}\\psi^{\\omega_i}_k$, keeping eigenvectors below a threshold $\\delta_\\lambda$ and multiplying them by partition-of-unity functions to form the interpolation operator $P$; the resulting two-grid method has convergence factor controlled by $K_{TG}=c_m H^2/\\Lambda^*$ from estimate (19). The paper reports that with this adaptive coarse space, PCG converges to a $10^{-9}$ residual in O(1) iterations across all tested contrasts, fracture counts, and matrix heterogeneities, while the implicit scheme remains first-order accurate in time.","pith_inferences":["Beyond the paper: the dominance condition could be monitored cheaply at runtime, and if the concentration approaches the assumed bounds, the upper-bound operator and local basis functions could be refreshed only in the affected subdomains.","Beyond the paper: the same additive-split-plus-spectral-coarse-space construction should transfer to other degenerate nonlinear parabolic problems with monotone coefficient dependence, such as unsaturated flow or gas-condensate transport, provided the local eigenvalue decay still separates the stiff modes.","Beyond the paper: the reported tables suggest that what drives the required number of basis functions is the number of small-eigenvalue modes created by fracture geometry rather than the permeability contrast itself; a quantitative formula linking coarse-space dimension to fracture density would be a natural next step."],"forward_implications":["Each nonlinear time step becomes one linear solve with the same matrix $A=S^{(\\mathrm{lin})}+\\tau D^{(\\mathrm{lin})}$, so spectral multiscale basis functions are computed once offline and reused for the entire simulation.","With the adaptive coarse space, PCG reaches a residual tolerance of $10^{-9}$ in O(1) iterations across contrasts $10^{3}$ to $10^{9}$, for 30 and 160 fractures, on both $10\\times10$ and $20\\times20$ coarse grids.","A single threshold $\\delta_\\lambda=10^{-3}$ automatically places more basis functions where fractures create small local eigenvalues, matching the performance of 4 to 8 fixed basis functions per node while keeping the coarse system smaller.","The linearly implicit scheme shows first-order time accuracy: relative $L^2$ errors shrink as the number of time steps increases in the reported tests."],"supporting_citations":[{"why":"Supplies the explicit-implicit-null idea of adding and subtracting a linear operator so stiff dynamics can be treated implicitly while the nonlinear residual is explicit.","marker":"[28]"},{"why":"Provides the two-grid convergence framework and the spectral equivalence between symmetric Gauss-Seidel smoothing and the matrix diagonal used in Section 4.3.","marker":"[76]"},{"why":"Defines the generalized multiscale finite element method whose local spectral basis construction the adaptive coarse space adapts.","marker":"[31]"},{"why":"Gives the earlier multiscale reduced-order construction for shale gas transport in fractured media that the adaptive spectral space extends.","marker":"[8]"},{"why":"Supplies the coupled discrete-fracture dual-continuum shale gas model and its coefficients, the physical starting point of the mixed-dimensional equations.","marker":"[4]"},{"why":"Introduces the adaptive two-grid preconditioner with a coarse space that the present work carries into the nonlinear mixed-dimensional setting.","marker":"[85]"},{"why":"Provides the kerogen-scale gas transport physics, including Langmuir adsorption and diffusion coefficients, that determine the nonlinear coefficients.","marker":"[6]"},{"why":"Establishes the two-grid error propagation and convergence estimates used to state the $K_{TG}=c_m H^2/\\Lambda^*$ bound.","marker":"[37]"},{"why":"Provides the canonical three-step representation of difference schemes that the appendix uses to prove unconditional stability.","marker":"[63]"}],"fun_headline_variants":["Adaptive spectral coarse space: O(1) solves for shale gas","Fixed linear split + adaptive coarse space tame high contrasts","O(1) PCG iterations for high-contrast fractured shale gas","Adaptive two-grid preconditioner yields robust shale gas solves"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that the fixed linear operator chosen as an upper bound over the concentration interval $[c_{\\min}, c_{\\max}]$ indeed dominates the true nonlinear operator at every time step; the paper sets the coefficients this way but never checks the resulting matrix inequalities during the simulations, and the upper-bound construction does not exactly match the positive-definiteness hypotheses stated in the stability theorem.","fun_headline_variants_meta":{"raw":{"variants":["Adaptive spectral coarse space: O(1) solves for shale gas","Fixed linear split + adaptive coarse space tame high contrasts","O(1) PCG iterations for high-contrast fractured shale gas","Adaptive two-grid preconditioner yields robust shale gas solves"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001184,"raw_usage":{"total_tokens":5016,"prompt_tokens":1201,"completion_tokens":3815,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":817,"completion_tokens_details":{"reasoning_tokens":3742}},"tokens_in":817,"tokens_out":3815,"duration_ms":25399,"temperature":1.0,"reasoning_tokens":3742,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:42:50.124985+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"At each time step of the reported 30- and 160-fracture runs, compute the smallest eigenvalue of $S^{(\\mathrm{lin})}-S^{(\\mathrm{nl})}(c^n)$ and of $D^{(\\mathrm{lin})}-D^{(\\mathrm{nl})}(c^n)$; if any eigenvalue is negative while $c^n$ lies inside $[c_{\\min}, c_{\\max}]$, the unconditional-stability claim fails. Separately, fix the threshold $\\delta_\\lambda=10^{-3}$ and increase the fracture-matrix permeability contrast beyond $10^{9}$; if the average number of PCG iterations rises systematically with contrast, the contrast-independence claim fails.","supporting_citations":[{"cited_title":"The explicit–implicit–null method: Removing the numerical insta- bility of pdes","cited_arxiv_id":null,"evidence_quote":"Supplies the explicit-implicit-null idea of adding and subtracting a linear operator so stiff dynamics can be treated implicitly while the nonlinear residual is explicit."},{"cited_title":"Multilevel block factorization preconditioners: Matrix-based analysis and algo- rithms for solving finite element equations","cited_arxiv_id":null,"evidence_quote":"Provides the two-grid convergence framework and the spectral equivalence between symmetric Gauss-Seidel smoothing and the matrix diagonal used in Section 4.3."},{"cited_title":"A two-grid preconditioner with an adaptive coarse space for flow simulations in highly heterogeneous media","cited_arxiv_id":null,"evidence_quote":"Introduces the adaptive two-grid preconditioner with a coarse space that the present work carries into the nonlinear mixed-dimensional setting."},{"cited_title":"On two-grid convergence estimates","cited_arxiv_id":null,"evidence_quote":"Establishes the two-grid error propagation and convergence estimates used to state the $K_{TG}=c_m H^2/\\Lambda^*$ bound."},{"cited_title":"The theory of difference schemes , volume 240","cited_arxiv_id":null,"evidence_quote":"Provides the canonical three-step representation of difference schemes that the appendix uses to prove unconditional stability."}],"review_version":1}