Pith. sign in

REVIEW 3 major objections 3 minor 40 references

AlgMortar: a fully algebraic multiscale mortar preconditioner

T0 review · 3 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read AlgMortar rebuilds the multiscale mortar method from the matrix alone: for SPD M-matrices it is provably well posed, and in large Darcy-flow tests it matches or beats algebraic multigrid.

desk verdict Algebraically sound, honestly reported, and worth refereeing — but the corner-point experiments are not covered by the paper's own theory. read the letter →

arxiv 2607.22905 v1 pith:ZZTZVWCD submitted 2026-07-24 math.NA cs.NA

Luan F. Santos , Fabricio S. Sousa , Roberto F. Ausas , Rafael T. Guiraldello , Felipe Pereira This is my paper · ORCID
classification math.NAcs.NA MSC 65F0865N5565Y0565F10
keywords algebraicmultiscalemethodmortarpreconditionerdomaindecompositionM-matrixconjugategradientDarcyflowcorner-pointgridbasisfunctions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

AlgMortar proposes to precondition large sparse symmetric positive definite linear systems using only the matrix itself. The paper's central claim is that the multiscale mortar mixed finite element method—normally defined through grids, coefficients, and geometry—can be rebuilt algebraically: graph partition the matrix, solve local Dirichlet-like problems, and couple them with an interface condition that enforces weak flux continuity. The authors prove that for SPD matrices with nonpositive off-diagonal entries (M-matrices) the resulting preconditioner is well posed and positive definite, with a preconditioned condition number bounded by $1 + 2 K_\Lambda$, and they show numerically that on 50–110 million unknown Darcy-flow systems it matches or beats algebraic multigrid in solution time, especially when a relaxed tolerance is acceptable.

What carries the argument

The load-bearing identity is $B^T = 4 W M$, where $W$ is the diagonal matrix of inter-subdomain entries $a_{kl}$, $M$ averages endpoint pressures on each graph edge, and $B$ assembles weighted interface contributions back into subdomains. Around it, the method uses graph partitioning to define subdomains, modified local matrices $A_i = A_{ii} - \sum D_{ij}$ that mimic Dirichlet problems, multiscale basis functions computed by local Cholesky solves, and a piecewise-constant mortar space on each interface. This system yields the interface operator $S_\lambda = -(W + 4 W M \hat{A}^{-1} M^T W)$, whose SPD-ness on the mortar space is what makes conjugate gradients applicable; the quadratic-form identity in Theorem 4.4

What would settle it

Take an SPD system from a two-point flux finite-volume discretization on a deliberately distorted corner-point grid, or add small positive off-diagonal couplings to an M-matrix, run AlgMortar, and check whether every local $A_i$ stays positive definite and whether PCG still converges in the predicted iteration range; a single local Cholesky failure or a preconditioned matrix with eigenvalues below zero would show the M-matrix hypothesis is load-bearing.

Watch

Extended reading notes

Core claim

On its own terms, AlgMortar is the claim that MMMFEM's geometric ingredients—subdomains, local Dirichlet problems, and the weak continuity of normal flux—can be replaced by algebraic counterparts read off an SPD matrix $A$. Subdomains are graph partitions; local operators $A_i$ are $A_{ii}$ minus sums of diagonal blocks formed from inter-subdomain columns; the interface coupling is assembled from the matrix entries $a_{kl}$ on graph edges, with incidence and averaging operators satisfying $B^T = 4 W M$. The paper proves the coupled system matrix $Q$ is positive definite when $A$ and $-W$ are SPD, so the interface system and the reduced pressure system are SPD and PCG applies; the exact-recovery statement is tha

Load-bearing premise

The proof requires the fine-grid matrix to be an M-matrix—symmetric positive definite with nonpositive off-diagonal entries—with strictly negative entries on every graph-interface edge, so the diagonal weight matrix $-W$ and the local modified operators $A_i$ are all positive definite; the corner-point-grid experiments in the paper never verify this sign condition, and TPFA on non-K-orthogonal corner-point grids can produce positive off-diagonal couplings.

Editorial extensions

If this is right

  • For any SPD matrix with nonpositive off-diagonal entries, the AlgMortar interface and reduced pressure systems are SPD, so PCG with AlgMortar is guaranteed to converge (Theorem 4.4 and Corollaries 4.5, 5.1).
  • The preconditioned condition number is bounded by 1 + 2 K_Lambda, so convergence is controlled by how well the piecewise-constant mortar space captures average pressures on graph interfaces (Theorem 5.6).
  • On uniform Cartesian grids with K = I, AlgMortar exactly reproduces MMMFEM, meaning the algebraic interface condition is the same weak flux-continuity condition (Remarks 3.2–3.4).
  • After one preconditioner application, residuals are nonzero only on partition interfaces, so inexpensive local smoothing can be applied without changing the parallel structure (Section 5.4).
  • On 10^7–10^8 unknown Darcy-flow systems, AlgMortar scales well and matches or beats algebraic multigrid in solution time on the largest cases, with the edge widening when the tolerance is relaxed (Tables 1, 3–5).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's largest corner-point results sit outside the proved hypothesis: the well-posedness theorem requires strictly negative inter-subdomain entries, yet the SPE11 matrices are never checked for positive off-diagonal couplings. Reporting the sign pattern of those matrices would show whether the flagship runs are covered by the proof.
  • Because the graph partition and local factorizations are reusable when only the right-hand side changes, AlgMortar may be especially attractive for repeated solves in time-dependent or multiphase simulation; the paper lists this as future work, not a present claim.
  • The spectral bound suggests a quantitative predictor: PCG iterations should track the best approximation of edge averages by the mortar space, so adaptive enrichment—adding basis functions where the algebraic projection error is large—could be guided by the same quantity K_Lambda.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper proposes AlgMortar, an algebraic multiscale mortar preconditioner for SPD systems. It constructs a domain decomposition from the matrix graph, defines local modified Dirichlet operators, and enforces a weak flux-continuity condition through an algebraic interface system. The authors prove well-posedness under the assumptions that the fine matrix A is SPD and the interface-weight matrix W (whose entries are the off-diagonal couplings on graph interfaces) is negative definite. They also derive a spectral bound kappa(M^{-1}A) <= 1 + 2 K_Lambda for the preconditioned system. Numerical experiments on large Darcy-flow systems, including SPE10 and SPE11 corner-point grids, compare AlgMortar with BoomerAMG and report competitive or faster times in several large cases.

Significance. If the theoretical claims hold, this is a useful contribution: a genuinely algebraic multiscale mortar preconditioner with self-contained proofs and a non-fitted spectral indicator. I re-derived the central identities (Prop. 3.1, Lemmas 4.2-4.3, identity (4.5), Theorem 4.4) and they are correct. The preconditioner exactly recovering MMMFEM on uniform Cartesian grids with K=I is a nice structural result. The numerical study is substantial, with weak and strong scaling on systems up to about 10^8 unknowns. However, the paper's main theoretical guarantee is conditional on an M-matrix-type sign condition, and the flagship irregular-grid experiments do not verify that condition; this gap is load-bearing for the conclusion that AlgMortar is effective on corner-point grids.

major comments (3)
  1. [§7.4 and Theorem 4.4] The SPE11 corner-point experiments are not covered by the proved theory. Theorem 4.4 requires A and -W to be SPD, which for a TPFA discretization means nonpositive off-diagonal entries (M-matrix) and strictly negative interface couplings. On non-K-orthogonal corner-point grids, TPFA can produce positive off-diagonal couplings. The paper never verifies that the assembled SPE11B/C matrices satisfy these conditions, yet the Conclusion states that AlgMortar 'remained effective on corner-point grids.' If an interface edge has a_kl > 0, then -W is not SPD, the quadratic form (4.4) loses its positivity argument, and, more critically, the local operators A_i = A_ii - sum_j D_ij (Eq. 3.32) can become indefinite, so the Cholesky factorization in Algorithm 5.1 can fail. The empirical claim therefore lies outside the theorem's hypotheses. Please either verify the sign conditions for the actual SPE11
  2. [§3.3 and Algorithm 5.1] The sentence 'If A is symmetric positive definite and the diagonal entries of W are negative, then each A_i is also symmetric positive definite' is asserted without proof at the point where it first appears. This property is needed for the Cholesky factorization used in Algorithm 5.1. It follows later from Lemma 4.3 and identity (4.5), but only under the joint hypotheses A SPD and -W SPD. The paper should state the argument explicitly in Section 3.3 and, more importantly, acknowledge that outside this class A_i may lose definiteness. In its current form, the setup phase has no fallback or check for this failure mode.
  3. [Introduction / Abstract] The paper sometimes describes AlgMortar as operating on 'an SPD matrix' or 'using only an SPD matrix,' which overstates the domain of the theoretical results. The well-posedness theorem and the spectral bound require the additional sign condition that -W be positive definite (equivalently, all graph-interface couplings be strictly negative). The abstract does state 'nonpositive off-diagonal entries,' but the introduction and several remarks would benefit from the same qualification to avoid implying that the proof covers all SPD matrices.
minor comments (3)
  1. [Table 1 header] The header 'AlgMortar AMGPCG-block Jacobi interface solver' appears corrupted; it should probably be split into 'AlgMortar' (with two interface-solver columns) and 'AMG'.
  2. [Section 7] Typo: 'Darcy flow proble' should be 'Darcy flow problem.'
  3. [Section 3.2.3] The phrase 'We assume that the diagonal entries of W ... are negative, as is the case for M-matrices' is clear but should explicitly state that equality to zero is excluded on edges of E, since an edge is only present when a_kl != 0; strict negativity is what makes -W positive definite.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: AlgMortar's proof chain is self-contained; self-citations are motivational or implementation-only.

full rationale

The central derivation is self-contained. Theorem 4.4 proves Q>0 directly from the definitions of A, W, M, Z and the stated hypotheses 'A and -W are SPD matrices'; the quadratic-form identity (4.4) reads (p,eta)^T Q (p,eta) = (1/4)p^T A p - (lambda-Mp)^T W(lambda-Mp), which is positive definite because -W is SPD. No fitted parameter or experimental quantity enters. Theorem 5.6's bound kappa(M^{-1}A) <= 1 + 2 K_Lambda is an identity-style estimate: K_Lambda is defined in Eq. (5.4) as the exact maximum of 4||(I-Pi_Lambda)Mp||^2_{-W}/||p||^2_A, and the bound is derived algebraically through Proposition 5.3 and Theorem 5.5 (the latter a standard error-propagation fact cited to external work [44]); it is not calibrated to observed iteration counts or timings. Remarks 3.2-3.4 ('when A arises from a global TPFA discretization on a uniform Cartesian grid with K=I, AlgMortar exactly recovers MMMFEM') are a design-consistency check: the algebraic local operators and weak flux condition are defined so that they coincide with MMMFEM in that special case. This recovery claim is not used as evidence for well-posedness or for the AMG comparisons, so it is not a circular prediction. Self-citations ([9,17,21]) appear for motivation, for SPE10 setup, and for using MUMPS; none is load-bearing in the proof. The paper explicitly assumes 'We assume that the diagonal entries of W, corresponding to intersubdomain couplings, are negative, as is the case for M-matrices'; the SPE11C corner-point experiments do not verify this condition, but that is a correctness/verification gap for the empirical claim, not a circular step. The score of 2 reflects only the presence of non-load-bearing self-citations; there is no reduction of a predicted result to a fitted input or self-citation chain.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on: (a) the M-matrix/Stieltjes structure of A (so -W is positive definite and the local Dirichlet-like operators A_i stay SPD), which the paper states but never checks for the corner-point systems it runs; (b) the standard TPFA-to-RT0 equivalence from the mixed-FE literature used to justify the MMMFEM-recovery property; (c) the unproved assertion that A_i is SPD, needed for the Cholesky factorizations in Algorithm 5.1; (d) the design conjecture that a piecewise-constant, one-DOF-per-interface mortar space keeps the spectral constant K_Lambda small, which the theory leaves open and the experiments never measure; and (e) benchmark-specific data fidelity. The hand-chosen hyperparameters (400 subdomains/proc, 15+15 block-Jacobi steps with 80 blocks, interface rtol 1e-2, mortar space) directly affect the reported wall-clock advantage. No new physical entities are postulated.

free parameters (4)
  • mortar coarse space = piecewise constants, one DOF per graph interface E_ij
    The interface space Lambda is chosen by hand (Section 5.5); both the spectral constant K_Lambda in Theorem 5.6 and the observed PCG iteration counts depend on its approximation quality. It is not derived from the matrix or problem data.
  • subdomains per MPI process = 400
    Fixed hyperparameter controlling local problem size and interface count (Section 7: 'AlgMortar uses 400 subdomains per MPI process'). Materially affects wall-clock results.
  • block-Jacobi smoothing profile = 15 pre + 15 post, 80 blocks per MPI process
    Chosen 'because it provided the lowest computational times among the strategies tested' (Sections 5.4 and 7); 30 smoothing sweeps per iteration are a significant fraction of the reported time budget.
  • interface solve tolerance = 10^-2 (PCG with block-Jacobi)
    Relaxed interface tolerance trades accuracy for speed; the paper shows outer iteration count is insensitive (Table 1) but total time depends on this choice (Section 7).
assumptions (5)
  • domain assumption A is SPD with nonpositive off-diagonal entries (Stieltjes/M-matrix) and every graph-edge coupling a_kl is strictly negative, so that the diagonal weight matrix W satisfies -W > 0
    Theorem 4.4 and Corollaries 4.5/5.1 require -W > 0, and Section 3.3 asserts A_i > 0 from the same structure. The paper calls this 'commonly arising' but never verifies it for the SPE11 corner-point systems in Section 7.4, where non-K-orthogonal TPFA can produce positive off-diagonal couplings.
  • standard math TPFA finite-volume discretization is equivalent up to quadrature to lowest-order Raviart-Thomas mixed FEM for diagonal K
    Invoked via refs [5,17] in Section 2.3 to justify the claim that AlgMortar 'exactly recovers MMMFEM' on uniform Cartesian grids with K=I (Remarks 3.2-3.4).
  • domain assumption Each local modified operator A_i = A_ii - sum_j D_ij remains SPD, so the Cholesky factorizations of Algorithm 5.1 are valid
    Asserted without proof at Eq. (3.32) in Section 3.3; it follows for M-matrices because A_i = A_ii + nonnegative diagonal, but the paper does not show this and the algorithms depend on it.
  • ad hoc to paper The one-DOF-per-interface piecewise-constant mortar space approximates average interface pressures well (K_Lambda small)
    The spectral bound (5.4) is vacuous if K_Lambda is large; the method's efficiency in Section 7 depends on this conjecture holding for METIS/ParMETIS partitions. The paper never measures K_Lambda.
  • domain assumption Benchmark data fidelity: SPE10/SPE11 grids and permeability fields (with L^2-projected upscaling of SPE10 permeabilities) are as described and representative
    Sections 7.2-7.4; the reported conclusions are benchmark-specific and depend on the generated grids and the OPM-based TPFA assembler.

how reviews work

0 comments
Cite this review

Pith. "Pith review of AlgMortar: a fully algebraic multiscale mortar preconditioner." pith.science (2026). https://pith.science/paper/ZZTZVWCD

@misc{pith2026260722905,
  author       = {Pith},
  title        = {Pith review of: AlgMortar: a fully algebraic multiscale mortar preconditioner},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZZTZVWCD}},
  note         = {Machine review of arXiv:2607.22905}
}
read the original abstract

The solution of large-scale symmetric positive definite linear systems arising from discretizations of second-order elliptic equations is challenging, especially in applications with highly heterogeneous coefficients, such as flow in porous media, which can lead to severely ill-conditioned systems. In this context, multiscale methods have recently been used to accelerate Krylov subspace methods, owing to their favorable parallel scalability. In this work, we present AlgMortar, a fully algebraic realization of the Multiscale Mortar Mixed Finite Element Method (MMMFEM). The method uses only information extracted from the fine-grid system matrix, which facilitates its implementation in existing solvers. AlgMortar uses graph partitioning to define a domain decomposition directly from the matrix graph. On each subdomain, it builds local linear systems that mimic Dirichlet problems, and couples the resulting local solutions through an algebraic interface condition that recovers the weak flux-continuity mechanism of MMMFEM. We prove that the method is well posed when the fine-grid matrix is symmetric positive definite and has nonpositive off-diagonal entries, a structure commonly arising from discretizations of elliptic problems. Numerical experiments on fine-grid linear systems arising from finite-volume discretizations of Darcy flow problems show that, when used as a preconditioner for the conjugate gradient method, the proposed approach exhibits good scalability and is competitive with state-of-the-art algebraic multigrid methods for challenging heterogeneous, high-contrast test cases, including highly irregular corner-point grids.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 1 canonical work pages

  1. [22]

    2025 , eprint=

    Multiscale Graph Reduction for Heterogeneous and Anisotropic Discrete Diffusion Processes , author=. 2025 , eprint=

    2025

  2. [1]

    George Karypis and Vipin Kumar , title =

  3. [2]

    2003 , address =

    Saad, Yousef , title =. 2003 , address =. doi:10.1137/1.9780898718003 , edition =

    doi:10.1137/1.9780898718003 2003

  4. [3]

    Carvalho, Péricles R. P. and Silva Carvalho, Pablo Giovanni and Fracalossi Rocha, Franciane and Guiraldello, Rafael and Ausas, Roberto Federico and Pereira, Felipe and Sousa, Fabricio S. , title =. 2025 , howpublished =

    2025

  5. [4]

    Applied Numerical Mathematics , volume =

    Boomer. Applied Numerical Mathematics , volume =. 2002 , issn =. doi:10.1016/S0168-9274(01)00115-5 , author =

    doi:10.1016/s0168-9274(01)00115-5 2002

  6. [5]

    Christie, M. A. and Blunt, M. J. , title =. SPE Reservoir Evaluation & Engineering , volume =. 2001 , doi =

    2001

  7. [6]

    Journal of Computational Physics , volume =

    A highly parallelized multiscale preconditioner for. Journal of Computational Physics , volume =. 2025 , issn =. doi:10.1016/j.jcp.2024.113603 , author =

    arXiv 2025

  8. [7]

    An Introduction to Reservoir Simulation Using MATLAB/GNU Octave: User Guide for the MATLAB Reservoir Simulation Toolbox (MRST) , publisher=

    Lie, Knut-Andreas , year=. An Introduction to Reservoir Simulation Using MATLAB/GNU Octave: User Guide for the MATLAB Reservoir Simulation Toolbox (MRST) , publisher=

Show all 40 references
  1. [8]

    and Yotov, Ivan , title =

    Arbogast, Todd and Pencheva, Gergina and Wheeler, Mary F. and Yotov, Ivan , title =. Multiscale Modeling & Simulation , volume =. 2007 , doi =

    2007

  2. [9]

    and Wheeler, Mary F

    Arbogast, Todd and Cowsar, Lawrence C. and Wheeler, Mary F. and Yotov, Ivan , title =. SIAM Journal on Numerical Analysis , volume =. 2000 , doi =

    2000

  3. [10]

    and Thomas, J

    Raviart, P.-A. and Thomas, J. M. , title =. Mathematical Aspects of the Finite Element Method , editor =. 1977 , address =

    1977

  4. [11]

    and Douglas, J

    Brezzi, F. and Douglas, J. Jr. and Marini, L. D. , title =. Numerische Mathematik , volume =. 1985 , pages =

    1985

  5. [12]

    Guiraldello and Roberto F

    Rafael T. Guiraldello and Roberto F. Ausas and Fabricio S. Sousa and Felipe Pereira and Gustavo C. Buscaglia , doi =. The. Journal of Computational Physics , volume =. 2018 , issn =

    2018

  6. [13]

    Implementation of a mortar mixed finite element method using a Multiscale Flux Basis , journal =

    Benjamin Ganis and Ivan Yotov , doi =. Implementation of a mortar mixed finite element method using a Multiscale Flux Basis , journal =. 2009 , issn =

    2009

  7. [14]

    Transport in Porous Media , year =

    Jaramillo, Alfredo and McCaskill, Bradley and Piri, Mohammad and Masalmeh, Shehadeh , title =. Transport in Porous Media , year =

  8. [15]

    Applied Mathematical Modelling , volume =

    An algebraic multiscale solver for the simulation of two-phase flow in heterogeneous and anisotropic porous media using general unstructured grids (. Applied Mathematical Modelling , volume =. 2022 , issn =. doi:10.1016/j.apm.2021.11.017 , author =

    2022 doi

  9. [16]

    Two-level mortar domain decomposition preconditioners for heterogeneous elliptic problems , journal =

    Todd Arbogast and Hailong Xiao , doi =. Two-level mortar domain decomposition preconditioners for heterogeneous elliptic problems , journal =. 2015 , issn =

    2015

  10. [17]

    2019 , issn =

    A two-grid preconditioner with an adaptive coarse space for flow simulations in highly heterogeneous media , journal =. 2019 , issn =. doi:10.1016/j.jcp.2019.03.038 , author =

    2019 doi

  11. [18]

    Bosma and Sergey Klevtsov and Olav Møyner and Nicola Castelletto , doi =

    Sebastian B.M. Bosma and Sergey Klevtsov and Olav Møyner and Nicola Castelletto , doi =. Enhanced multiscale restriction-smoothed basis (. Journal of Computational Physics , volume =. 2021 , issn =

    2021

  12. [19]

    2025 , issn =

    Multiscale approximation and two-grid preconditioner for extremely anisotropic heat flow , journal =. 2025 , issn =. doi:10.1016/j.jcp.2025.114201 , author =

    2025

  13. [20]

    SIAM Journal on Scientific Computing , volume =

    Fu, Shubin and Chung, Eric and Zhao, Lina , title =. SIAM Journal on Scientific Computing , volume =. 2024 , doi =

    2024

  14. [21]

    An algebraic multiscale preconditioner for large sparse

    Yingjie Zhou and Shubin Fu and Eric Tsz Shun Chung , year=. An algebraic multiscale preconditioner for large sparse. 2606.04864 , archivePrefix=

    arXiv

  15. [23]

    and Paz, Stevens and Ausas, Roberto F

    Jaramillo, Alfredo and Guiraldello, Rafael T. and Paz, Stevens and Ausas, Roberto F. and Sousa, Fabricio S. and Pereira, Felipe and Buscaglia, Gustavo C. , title =. Computational Geosciences , volume =. 2022 , doi =

    2022

  16. [24]

    and Abhyankar, S

    Balay, S. and Abhyankar, S. and Adams, M. F. and Brown, J. and Brune, P. and Buschelman, K. and Constantinescu, E. M. and Dalcin, L. and Benson, S. and Dener, A. and others , title =. 2025 , month =

    2025

  17. [25]

    and Duff, Iain S

    Amestoy, Patrick R. and Duff, Iain S. and L'Excellent, Jean-Yves and Koster, Jacko , title =. SIAM Journal on Matrix Analysis and Applications , volume =. 2001 , doi =

    2001

  18. [26]

    2001 , note =

    A review of algebraic multigrid , journal =. 2001 , note =. doi:10.1016/S0377-0427(00)00516-1 , author =

    2001 doi

  19. [27]

    2013 , note =

    Karypis, George and Schloegel, Kirk , institution =. 2013 , note =

    2013

  20. [28]

    Mathematics and Computers in Simulation , volume =

    Interface spaces for the. Mathematics and Computers in Simulation , volume =. 2019 , issn =

    2019

  21. [29]

    2025 , issn =

    Multiscale mixed methods with improved accuracy: The role of oversampling and smoothing , journal =. 2025 , issn =

    2025

  22. [30]

    2026 , issn =

    Fast converging parallel offline–online iterative multiscale mixed methods , journal =. 2026 , issn =

    2026

  23. [31]

    Chung and Shubin Fu and Yanfang Yang , title =

    Eric T. Chung and Shubin Fu and Yanfang Yang , title =. Communications in Computational Physics , year=

  24. [32]

    A multiscale two-point flux-approximation method , journal =

    Olav Møyner and Knut-Andreas Lie , doi =. A multiscale two-point flux-approximation method , journal =. 2014 , issn =

    2014

  25. [33]

    Algorithmic monotone multiscale finite volume methods for porous media flow , journal =

    Omar Chaabi and Mohammed. Algorithmic monotone multiscale finite volume methods for porous media flow , journal =. 2024 , issn =

    2024

  26. [34]

    2000 , issn =

    Multigrid on the interface for mortar mixed finite element methods for elliptic problems , journal =. 2000 , issn =

    2000

  27. [35]

    SIAM Journal on Scientific Computing , volume =

    Vecharynski, Eugene and Saad, Yousef and Sosonkina, Masha , title =. SIAM Journal on Scientific Computing , volume =. 2014 , doi =

    2014

  28. [36]

    2015 , address =

    Dolean, Victorita and Jolivet, Pierre and Nataf, Frédéric , title =. 2015 , address =

    2015

  29. [37]

    and Yotov, Ivan , title =

    Arbogast, Todd and Wheeler, Mary F. and Yotov, Ivan , title =. SIAM Journal on Numerical Analysis , volume =. 1997 , doi =

    1997

  30. [38]

    and Huang, Jizu , title =

    Ye, Changqing and Fu, Shubin and Chung, Eric T. and Huang, Jizu , title =. SIAM Journal on Scientific Computing , volume =. 2024 , doi =

    2024

  31. [39]

    and Pereira, Felipe and Sousa, Fabr

    Zhou, Dilong and Guiraldello, Rafael T. and Pereira, Felipe and Sousa, Fabr. Accelerating. 2026 , eprint =

    2026

  32. [40]

    2026 , eprint =

    Graph-Based Meshfree Multi-scale Coarse Space Approximation for Two-Level Schwarz Methods , author =. 2026 , eprint =

    2026

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.