pith. sign in

arxiv: 2604.05315 · v1 · submitted 2026-04-07 · 🧮 math.NA · cs.NA· math.AP

Higher-Order Multiscale Computational Method for Multi-Continuum Problems in Highly Heterogeneous Media

Hao Dong , Jiayuan Peng , Jian Huang This is my paper

Pith reviewed 2026-05-10 19:54 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.AP
keywords higher-order multiscale methodmulti-continuum problemshighly heterogeneous mediahomogenizationasymptotic analysisfinite element methodconvergence ratenumerical algorithm
0
0 comments X

The pith

A higher-order multiscale asymptotic solution derived from unit cell functions approximates multi-continuum problems in highly heterogeneous media with proven integral convergence.

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

The authors define microscopic unit cell functions to derive macroscopic homogenized equations and formulas for effective parameters. This construction produces a higher-order multi-scale asymptotic solution whose pointwise approximation properties are analyzed and whose convergence rate in the integral norm is rigorously established under certain assumptions. A numerical algorithm combining the finite element method, finite difference method, and interpolation is implemented, and experiments confirm its accuracy, efficiency, and stability.

Core claim

Defining microscopic unit cell functions yields homogenized macroscopic equations and effective parameters that form an HOMS asymptotic solution; the solution approximates the original multi-continuum equations pointwise and converges at a specific rate in the integral norm under the given assumptions on heterogeneity and scale separation.

What carries the argument

Higher-order multi-scale (HOMS) asymptotic solution obtained from microscopic unit cell functions to produce homogenized equations and effective parameters.

If this is right

  • Solutions to multi-continuum problems in highly heterogeneous media can be computed at macroscopic scale while retaining controlled approximation error.
  • The combined finite-element, finite-difference, and interpolation algorithm provides a stable practical implementation with documented high accuracy.
  • Error bounds from the integral-norm convergence rate supply a priori guarantees on solution quality.
  • The method remains efficient even when the underlying heterogeneity is fine-scale and complex.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same unit-cell homogenization strategy could be tested on time-dependent or nonlinear multi-continuum models if the scale-separation assumption continues to hold.
  • Adaptive selection of unit-cell resolution based on local heterogeneity strength might further reduce computational cost in domains with spatially varying scale separation.
  • The approach could be paired with existing upscaling techniques for porous-media flow or composite materials to obtain similar error guarantees.

Load-bearing premise

The media must exhibit sufficient scale separation and uniform heterogeneity properties for the unit cell derivation and convergence analysis to remain valid.

What would settle it

Numerical tests in which the observed integral-norm error fails to decrease at the predicted rate when the scale separation parameter is systematically reduced would disprove the convergence claim.

Figures

Figures reproduced from arXiv: 2604.05315 by Hao Dong, Jian Huang, Jiayuan Peng.

Figure 1
Figure 1. Figure 1: Structure of sandstone (from Zhao et al. [1]) ∗Corresponding author. hao.dong@xidian.edu.cn (H. Dong) ORCID(s): Some of the first studies on porous media treated the porous medium as a single continuum. Fluid flow in porous media is often described at the macroscopic scale by Darcy’s Law, which assigns effective properties to each representa￾tive volume through local simulations to construct coarse￾grid eq… view at source ↗
Figure 2
Figure 2. Figure 2: , where 𝜀 = 1∕8. The input parameters for the validation example are given in [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 5
Figure 5. Figure 5: Evolution of the relative errors of the 𝐿2 norm and 𝐻1 seminorm over time: (a) Lerr1; (b) Herr1; (c) Lerr2; (d) Herr2. requirements for engineering computations. Furthermore, as shown in [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: The pressure field at 𝑡 = 1.0 of second-continuum media [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 7
Figure 7. Figure 7: and [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The pressure field in 𝑥3 = 0.5 at 𝑡 = 1.0 of second￾continuum media. 5.3. Example 3. 2D channel media For the two-dimensional case of problem (3), consider the macroscopic domain Ω = (𝑥1 , 𝑥2 ) = [0, 1]2 and the microscopic unit cell 𝑌 = (𝑦1 , 𝑦2 ) = [0, 1]2 as shown in [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Evolution of the relative errors of the 𝐿2 norm and 𝐻1 seminorm over time: (a) Lerr1; (b) Herr1; (c) Lerr2; (d) Herr2 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The schematic of 2D periodic media [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: and [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The pressure field at 𝑡 = 1.0 of second-continuum media. (a) (b) (c) (d) [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Evolution of the relative errors of the 𝐿2 norm and 𝐻1 seminorm over time: (a) Lerr1; (b) Herr1; (c) Lerr2; (d) Herr2. norm, the relative errors of 𝑢 (2𝜀) 1 and 𝑢 (2𝜀) 2 are only about 7% and 5%, respectively; under the 𝐻1 seminorm, the relative errors of 𝑢 (2𝜀) 1 and 𝑢 (2𝜀) 2 are only approximately 10% and 8%, significantly lower than those of the homogenized solution and the FOMS solution, thus meeting … view at source ↗
Figure 16
Figure 16. Figure 16: The pressure field in 𝑥3 = 0.375 at 𝑡 = 1.0 of second￾continuum media. (a) (b) (c) (d) [PITH_FULL_IMAGE:figures/full_fig_p016_16.png] view at source ↗
Figure 15
Figure 15. Figure 15: The pressure field in 𝑥3 = 0.375 at 𝑡 = 1.0 of first￾continuum media. the HOMS method is significantly smaller than that of the direct FEM, and its efficiency is also markedly superior to the FEM. As shown in [PITH_FULL_IMAGE:figures/full_fig_p016_15.png] view at source ↗
Figure 17
Figure 17. Figure 17: , the HOMS method exhibits excellent numerical stability over time and can be effectively applied to the computation of dynamic problem (3) that evolves with time. 6. Summary and Outlook 6.1. Summary of Research Findings This paper investigates multi-continuum problems in highly heterogeneous media, and develops corresponding (a) 𝑢 (0) 2 (b) 𝑢 (1𝜀) 2 (c) 𝑢 (2𝜀) 2 (d) 𝑢 𝜀 2 [PITH_FULL_IMAGE:figures/full_f… view at source ↗
read the original abstract

This paper presents a high-accuracy higher-order multiscale method for solving multi-continuum problems in in highly heterogeneous media. First, microscopic unit cell functions are defined, leading to the derivation of macroscopic homogenized equations and formulas for calculating effective parameters, which yield a higher-order multi-scale (HOMS) asymptotic solution. Subsequently, the pointwise approximation properties of this solution to the original equations are analyzed, and its convergence rate in the integral norm is rigorously established under certain assumptions. Furthermore, a multiscale numerical algorithm is developed by integrating the finite element method (FEM), finite difference method, and interpolation technique. Finally, numerical experiments demonstrate the high accuracy, efficiency, and stability of the proposed HOMS numerical algorithm.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper develops a higher-order multiscale (HOMS) method for multi-continuum problems in highly heterogeneous media. Microscopic unit-cell problems are solved to derive a homogenized macroscopic multi-continuum system together with explicit formulas for the effective coefficients; a higher-order asymptotic expansion is constructed, its pointwise approximation properties are analyzed, and a convergence rate in the integral norm is proved under assumptions on scale separation and media heterogeneity. A hybrid numerical algorithm combining FEM, FDM and interpolation is assembled, and numerical experiments are presented to illustrate accuracy, efficiency and stability.

Significance. If the stated convergence analysis is correct, the work supplies a concrete higher-order extension of classical homogenization that retains rigorous error control while remaining computationally tractable. The explicit construction of effective parameters from independent unit-cell problems and the hybrid solver constitute reusable methodological contributions for applications in porous-media flow, composite materials and other multi-continuum settings.

minor comments (3)
  1. The abstract and introduction state that convergence holds “under certain assumptions,” but the precise hypotheses (e.g., bounds on the contrast, minimal scale-separation ratio, regularity of the coefficients) should be collected in a single, prominently placed statement early in the paper so that readers can immediately assess applicability.
  2. In the numerical section, error tables or plots versus the small-scale parameter ε would make the observed convergence rates directly comparable to the theoretical prediction; currently the experiments focus on absolute accuracy without systematic variation of ε.
  3. Notation for the multi-continuum variables (e.g., the distinction between microscopic and macroscopic continua) is introduced gradually; a short table or diagram summarizing the symbols and their domains would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our work on the higher-order multiscale (HOMS) method. The recommendation of minor revision is noted. However, the report contains no specific major comments to address.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines independent microscopic unit-cell problems to derive the macroscopic homogenized multi-continuum equations and effective coefficients, then proves pointwise approximation properties and an integral-norm convergence rate for the resulting higher-order asymptotic solution under explicit assumptions on scale separation and heterogeneity. The numerical algorithm is assembled from standard FEM/FDM/interpolation discretizations of the derived system. No load-bearing step reduces the claimed convergence result or effective parameters to a self-definition, a fitted input renamed as a prediction, or a self-citation chain; all analytic statements remain conditioned on external assumptions rather than being tautological by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the existence of well-defined microscopic unit cell problems that yield effective parameters and on unspecified assumptions that enable the higher-order asymptotic expansion and convergence proof. No explicit free parameters or invented entities are named in the abstract.

axioms (1)
  • domain assumption Certain assumptions on the media heterogeneity and scale separation allow the higher-order asymptotic expansion and convergence in the integral norm.
    Invoked in the abstract when stating that the convergence rate is rigorously established under certain assumptions.

pith-pipeline@v0.9.0 · 5421 in / 1360 out tokens · 35486 ms · 2026-05-10T19:54:08.649919+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages

  1. [1]

    Understanding stress-sensitivebehaviorofporestructureintightsandstonereservoirs under cyclic compression using mineral, morphology, and stress analyses,

    N. Zhao, L. Wang, L. Sima, Y. Guo, and H. Zhang, “Understanding stress-sensitivebehaviorofporestructureintightsandstonereservoirs under cyclic compression using mineral, morphology, and stress analyses,”JournalofPetroleumScienceandEngineering218,110987 (2022)

    work page 2022

  2. [2]

    The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains,

    P. Henning and M. Ohlberger, “The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains,” NUMERISCHE MATHEMATIK113, 601–629 (2009)

    work page 2009

  3. [3]

    On the Homogenization of the Stokes Problem in a Perforated Domain,

    M. Hillairet, “On the Homogenization of the Stokes Problem in a Perforated Domain,” ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS230, 1179–1228 (2018)

    work page 2018

  4. [4]

    Homogenization of the navier-stokes equations in open sets perforated with tiny holes i. abstract framework, a volume distri- butionofholes,

    G. Allaire, “Homogenization of the navier-stokes equations in open sets perforated with tiny holes i. abstract framework, a volume distri- butionofholes,”ArchiveforRationalMechanics&Analysis (1991)

    work page 1991

  5. [5]

    G. Allaire, “Homogenization of the navier-stokes equations in open sets perforated with tiny holes ii: Non-critical sizes of the holes for a volume distribution and a surface distribution of holes,” Archive for Rational Mechanics & Analysis (1991)

    work page 1991

  6. [6]

    Homogenization of the navier-stokes equations with a slip boundary condition,

    G. Allaire, “Homogenization of the navier-stokes equations with a slip boundary condition,” Communications on Pure and Applied Mathematics (1991)

    work page 1991

  7. [7]

    Generalized multiscale finite element methods (gmsfem),

    Y. Efendiev, J. Galvis, and T. Y. Hou, “Generalized multiscale finite element methods (gmsfem),” Journal of Computational Physics251, 116–135 (2013)

    work page 2013

  8. [8]

    A conservative local multiscale model reduction technique for Stokes flows in hetero- geneous perforated domains,

    E. T. Chung, M. Vasilyeva, and Y. Wang, “A conservative local multiscale model reduction technique for Stokes flows in hetero- geneous perforated domains,” JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS321, 389–405 (2017)

    work page 2017

  9. [9]

    Generalized multiscale finite element methods for problems in perforated het- erogeneous domains,

    E. T. Chung, Y. Efendiev, G. Li, and M. Vasilyeva, “Generalized multiscale finite element methods for problems in perforated het- erogeneous domains,” APPLICABLE ANALYSIS95, 2254–2279 (2016)

    work page 2016

  10. [10]

    Mixed GMsFEM for secondorderellipticprobleminperforateddomains,

    E. T. Chung, W. T. Leung, and M. Vasilyeva, “Mixed GMsFEM for secondorderellipticprobleminperforateddomains,”JOURNALOF COMPUTATIONAL AND APPLIED MATHEMATICS304, 84–99 (2016)

    work page 2016

  11. [11]

    Online adaptive local multiscale model reduction for heterogeneous problems in perforated domains,

    E. T. Chung, Y. Efendiev, W. T. Leung, M. Vasilyeva, and Y. Wang, “Online adaptive local multiscale model reduction for heterogeneous problems in perforated domains,” APPLICABLE ANALYSIS96, 2002–2031 (2017)

    work page 2002

  12. [12]

    On time integrators for GeneralizedMultiscaleFiniteElementMethodsappliedtoadvection- diffusion in high-contrast multiscale media,

    W. Xie, J. Galvis, Y. Yang, and Y. Huang, “On time integrators for GeneralizedMultiscaleFiniteElementMethodsappliedtoadvection- diffusion in high-contrast multiscale media,” JOURNAL OF COM- PUTATIONAL AND APPLIED MATHEMATICS460(2025)

    work page 2025

  13. [13]

    Hierarchicalmultiscalefiniteelement method for multi-continuum media,

    J.S.R.ParkandV.H.Hoang,“Hierarchicalmultiscalefiniteelement method for multi-continuum media,” JOURNAL OF COMPUTA- TIONAL AND APPLIED MATHEMATICS369(2020)

    work page 2020

  14. [14]

    Homogenization of a multiscale multi-continuum system,

    J. S. R. Park and V. H. Hoang, “Homogenization of a multiscale multi-continuum system,” APPLICABLE ANALYSIS101, 1271– 1298 (2022)

    work page 2022

  15. [15]

    Multiscale simulations for upscaled multi-continuum flows,

    J. S. R. Park, S. W. Cheung, T. Mai, and V. H. Hoang, “Multiscale simulations for upscaled multi-continuum flows,” JOURNAL OF COMPUTATIONALANDAPPLIEDMATHEMATICS374(2020)

    work page 2020

  16. [16]

    Multiscale simulations for multi-continuum Richards equations,

    J. S. R. Park, S. W. Cheung, and T. Mai, “Multiscale simulations for multi-continuum Richards equations,” JOURNAL OF COMPUTA- TIONAL AND APPLIED MATHEMATICS397(2021)

    work page 2021

  17. [17]

    Basic concepts in the theory of seepage ofhomogeneous liquids in fissured rocks,

    G. I. Barenblatt, Y. P. Zheltov, and I. N. Kochina, “Basic concepts in the theory of seepage ofhomogeneous liquids in fissured rocks,” Journal of Applied Mathematics24, 5–1303 (1960)

    work page 1960

  18. [18]

    Coupling ofmultiscaleandmulti-continuumapproaches,

    E. T. Chung, Y. Efendiev, T. Leung, and M. Vasilyeva, “Coupling ofmultiscaleandmulti-continuumapproaches,”GEM-International Journal on Geomathematics8, 9–41 (2017)

    work page 2017

  19. [19]

    On fluid reserves and the produc- tion of superheated steam from fractured, vapor-dominated geother- mal reservoirs,

    K. Pruess and T. N. Narasimhan, “On fluid reserves and the produc- tion of superheated steam from fractured, vapor-dominated geother- mal reservoirs,” Journal of Geophysical Research87, 9329 (1982)

    work page 1982

  20. [20]

    Practical method for modeling fluid and heat flow in fractured porous media,

    K. Pruess and T. N. Narasimhan, “Practical method for modeling fluid and heat flow in fractured porous media,” Society of Petroleum Engineers Journal25, 14–26 (1985)

    work page 1985

  21. [21]

    A multiple-porosity method for simulation ofnaturallyfracturedpetroleumreservoirs,

    Y. S. Wu and K. Pruess, “A multiple-porosity method for simulation ofnaturallyfracturedpetroleumreservoirs,”SpeReservoirEngineer- ing3, 327–336 (1988)

    work page 1988

  22. [22]

    Mul- ticontinuum homogenization in perforated domains,

    W. Xie, Y. Efendiev, Y. Huang, W. T. Leung, and Y. Yang, “Mul- ticontinuum homogenization in perforated domains,” JOURNAL OF COMPUTATIONAL PHYSICS530(2025)

    work page 2025

  23. [23]

    Multicontinuum homogenization for coupled flow and transport equations,

    D.Ammosov,J.Huang,W.T.Leung, andB.Shan,“Multicontinuum homogenization for coupled flow and transport equations,” JOUR- NAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 471(2026)

    work page 2026

  24. [24]

    Generalized multiscale finiteelementmethodformulticontinuumcoupledflowandtransport model,

    A. D. A., H. J., L. W. T., and S. B., “Generalized multiscale finiteelementmethodformulticontinuumcoupledflowandtransport model,” Lobachevskii journal of mathematics , 45 (2024)

    work page 2024

  25. [25]

    A multiple- continuum model for simulating single-phase and multiphase flow in naturally fractured vuggy reservoirs,

    Y.-S. Wu, Y. Di, Z. Kang, and P. Fakcharoenphol, “A multiple- continuum model for simulating single-phase and multiphase flow in naturally fractured vuggy reservoirs,” Journal of Petroleum Science and Engineering78, 13–22 (2011)

    work page 2011

  26. [26]

    Amulti-continuummultipleflowmech- anism simulator for unconventional oil and gas recovery,

    X.Li,D.Zhang, andS.Li,“Amulti-continuummultipleflowmech- anism simulator for unconventional oil and gas recovery,” Journal of Natural Gas Science and Engineering26, 652–669 (2015)

    work page 2015

  27. [27]

    Nu- mericalsimulationofwater-oilflowinnaturallyfracturedreservoirs,

    H. Kazemi, L. S. Merrill, K. L. Porterfield, and P. R. Zeman, “Nu- mericalsimulationofwater-oilflowinnaturallyfracturedreservoirs,” Society of Petroleum Engineers Journal16, 317–326 (1976)

    work page 1976

  28. [28]

    Thebehaviorofnaturallyfracturedreser- voirs,

    J.E.WarrenandP.J.Root,“Thebehaviorofnaturallyfracturedreser- voirs,” Society of Petroleum Engineers Journal3, 245–255 (1963)

    work page 1963

  29. [29]

    Bensoussan and Alain,Asymptotic analysis for periodic structures / (Asymptotic analysis for periodic structures /, 1978)

    work page 1978

  30. [30]

    Mathematical problems in elasticity and homogenization,

    O. Oleinik, A. Shamaev, and G. Yosifian, “Mathematical problems in elasticity and homogenization,” (2012)

    work page 2012

  31. [31]

    An introduction to homogenization,

    D. Cioranescu and P. Donato, “An introduction to homogenization,” Oxford University Press, (1999)

    work page 1999

  32. [32]

    MULTISCALE COMPUTATION ANDCONVERGENCEFORCOUPLEDTHERMOELASTICSYS- TEM IN COMPOSITE MATERIALS,

    X. Wang, L. Cao, and Y. Wong, “MULTISCALE COMPUTATION ANDCONVERGENCEFORCOUPLEDTHERMOELASTICSYS- TEM IN COMPOSITE MATERIALS,” MULTISCALE MODEL- ING & SIMULATION13, 661–690 (2015)

    work page 2015

  33. [33]

    Multiscalecomputa- tional method for heat conduction problems of composite structures withdiverseperiodicconfigurationsindifferentsubdomains,

    H.Dong,J.Cui,Y.Nie,Z.Yang, andZ.Yang,“Multiscalecomputa- tional method for heat conduction problems of composite structures withdiverseperiodicconfigurationsindifferentsubdomains,”COM- PUTERS & MATHEMATICS WITH APPLICATIONS76, 2549– 2565 (2018)

    work page 2018

  34. [34]

    Multiscaleasymptoticexpansionandfiniteelementmethods for the mixed boundary value problems of second order elliptic equation in perforated domains,

    L.Cao,“Multiscaleasymptoticexpansionandfiniteelementmethods for the mixed boundary value problems of second order elliptic equation in perforated domains,” NUMERISCHE MATHEMATIK 103, 11–45 (2006)

    work page 2006

  35. [35]

    Second-ordertwo-scalecom- putationalmethodfordampeddynamicthermo-mechanicalproblems of quasi-periodic composite materials,

    H.Dong,J.Cui,Y.Nie, andZ.Yang,“Second-ordertwo-scalecom- putationalmethodfordampeddynamicthermo-mechanicalproblems of quasi-periodic composite materials,” JOURNAL OF COMPUTA- TIONAL AND APPLIED MATHEMATICS343, 575–601 (2018). :Preprint submitted to Elsevier Page 19 of 20

    work page 2018

  36. [36]

    SECOND-ORDER TWO-SCALE ANALYSIS METHOD FOR DYNAMIC THERMO- MECHANICAL PROBLEMS OF COMPOSITE STRUCTURES WITH CYLINDRICAL PERIODICITY,

    H. Dong, Y. Nie, J. Cui, Z. Yang, and Z. Wang, “SECOND-ORDER TWO-SCALE ANALYSIS METHOD FOR DYNAMIC THERMO- MECHANICAL PROBLEMS OF COMPOSITE STRUCTURES WITH CYLINDRICAL PERIODICITY,” INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING15, 834–863 (2018)

    work page 2018

  37. [37]

    Multiscale nu- mericalalgorithmsforelasticwaveequationswithrapidlyoscillating coefficients,

    Q.-l. Dong, L.-q. Cao, X. Wang, and J.-z. Huang, “Multiscale nu- mericalalgorithmsforelasticwaveequationswithrapidlyoscillating coefficients,” APPLIED MATHEMATICS AND COMPUTATION 336, 16–35 (2018)

    work page 2018

  38. [38]

    Asymptoticexpansionsandnumericalalgorithms of eigenvalues and eigenfunctions of the Dirichlet problem for sec- ond order elliptic equations in perforated domains,

    L.CaoandJ.Cui,“Asymptoticexpansionsandnumericalalgorithms of eigenvalues and eigenfunctions of the Dirichlet problem for sec- ond order elliptic equations in perforated domains,” NUMERISCHE MATHEMATIK96, 525–581 (2004)

    work page 2004

  39. [39]

    Multi-scale analysis and FE computation for the structure of composite materials with small periodic configuration under condition of coupled thermoelasticity,

    Y. Feng and J. Cui, “Multi-scale analysis and FE computation for the structure of composite materials with small periodic configuration under condition of coupled thermoelasticity,” INTERNATIONAL JOURNALFORNUMERICALMETHODSINENGINEERING60, 1879–1910 (2004)

    work page 1910

  40. [40]

    Multiscale asymptotic expansions and numerical algorithms for the wave equations of second order with rapidly oscillating coefficients,

    Q.-L. Dong and L.-Q. Cao, “Multiscale asymptotic expansions and numerical algorithms for the wave equations of second order with rapidly oscillating coefficients,” APPLIED NUMERICAL MATHE- MATICS59, 3008–3032 (2009)

    work page 2009

  41. [41]

    Multiscaleasymptoticexpansionsmeth- ods and numerical algorithms for the wave equations in perforated domains,

    Q.-L.DongandL.-Q.Cao,“Multiscaleasymptoticexpansionsmeth- ods and numerical algorithms for the wave equations in perforated domains,”APPLIEDMATHEMATICSANDCOMPUTATION232, 872–887 (2014). :Preprint submitted to Elsevier Page 20 of 20

    work page 2014