Finite difference heterogeneous multi-scale method for homogenization problems. (English) Zbl 1034.65067

Authors’ abstract: We propose a numerical method, the finite difference heterogeneous multi-scale method, for solving multi-scale parabolic problems. Based on the framework introduced by W. E. and B. Engquist [The heterogeneous multi-scale methods, Commun. Math. Sci. 1, 87–132 (2003)], the numerical method relies on the use of two different schemes for the original equation, at different grid level which allows to give numerical results at a much lower cost than solving the original equations. We describe the strategy for constructing such a method, discuss generalization for cases with time dependency, random correlated coefficients, non-conservative form and implementation issues. Finally, the new method is illustrated with several test examples.


65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs


Full Text: DOI


[1] Abdulle, A., Fourth order Chebyshev methods with recurrence relation, SIAM J. sci. comput., 23, 6, 2041-2054, (2002) · Zbl 1009.65048
[2] Abdulle, A.; Medovikov, A.A., Second order Chebyshev methods based on orthogonal polynomials, Numer. math., 90, 1, 1-18, (2001) · Zbl 0997.65094
[3] Allaire, G., Homogenization and two-scale convergence, SIAM J. math. anal., 23, 6, 1482-1518, (1992) · Zbl 0770.35005
[4] I. Babuska, Homogenization and its applications, in: B. Hubbard (Ed.), SYNSPADE, 1975, pp. 89-116
[5] Bensoussan, A.; Lions, J.-L.; Papanicolaou, G., Asymptotic analysis for periodic structures, (1978), North-Holland Amsterdam · Zbl 0411.60078
[6] Cioranescu, D.; Donato, P., An introduction to homogenization, (1999), Oxford University Press Oxford · Zbl 0939.35001
[7] E, W., Homogenization of linear and nonlinear transport equations, Commun. pure appl., XLV, 301-326, (1992) · Zbl 0794.35014
[8] E, W.; Engquist, B., The heterogeneous multi-scale methods, Comm. math. sci., 1, 1, 87-132, (2003) · Zbl 1093.35012
[9] Engquist, B., Computation of oscillatory solutions to hyperbolic differential equations, Springer lecture notes math., 1270, 10-22, (1987)
[10] Dorobantu, M.; Engquist, B., Wavelets-based numerical homogenization, SIAM J. numer. anal., 35, 2, 540-559, (1998) · Zbl 0936.65135
[11] Engquist, B.; Runborg, O., Wavelets-based numerical homogenization with applications, (), 97-148 · Zbl 0989.65117
[12] Hairer, E.; Nørsett, S.P.; Wanner, G., Solving ordinary differential equations I. nonstiff problems, () · Zbl 0638.65058
[13] Hairer, E.; Wanner, G., Solving ordinary differential equations II. stiff and differential-algebraic problems, () · Zbl 0729.65051
[14] Hou, T.-Y.; Wu, X.-H., A multiscale finite element method for elliptic problems in composite materials and porous media, J. comput. phys., 134, 169-189, (1997) · Zbl 0880.73065
[15] Hou, T.-Y.; Wu, X.-H.; Cai, Z., Convergence of a multi-scale finite element method for elliptic problems with rapidly oscillating coefficients, Math. comput., 68, 227, 913-943, (1999) · Zbl 0922.65071
[16] J.L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications, vols. 1 and 2, Paris, Dunod, 1968 (English Transl., Springer, 1970)
[17] Nguetseng, G., A general convergence result for a functional related to the theory of homogenization, SIAM J. math. anal., 20, 3, 608-623, (1989) · Zbl 0688.35007
[18] Matache, A-M.; Babuska, I.; Schwab, C., Generalized p-FEM in homogenization, Numer. math., 86, 2, 319-375, (2000) · Zbl 0964.65125
[19] Schwab, C.; Matache, A.-M., Generalized FEM for homogenization problems, (), 197-238
[20] Neuss, N.; Jäger, W.; Wittum, G., Homogenization and multigrid, Computing, 66, 1, 1-26, (2001) · Zbl 0992.35013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.