메뉴 건너뛰기




Volumn 200, Issue 37-40, 2011, Pages 2710-2726

Adaptive finite element heterogeneous multiscale method for homogenization problems

Author keywords

A posteriori error estimate; Adaptive mesh refinement; Finite element method; Heterogenenous multiscale method; Homogenization; Multiscale method

Indexed keywords

ADAPTIVE MESH REFINEMENT; FINITE ELEMENT; HOMOGENIZATION; MULTISCALE METHOD; POSTERIORI ERROR ESTIMATES;

EID: 79958714011     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2010.06.012     Document Type: Article
Times cited : (57)

References (44)
  • 1
    • 33645988923 scopus 로고    scopus 로고
    • On a-priori error analysis of fully discrete heterogeneous multiscale FEM
    • Abdulle A. On a-priori error analysis of fully discrete heterogeneous multiscale FEM. SIAM Multiscale Model. Simul. 2005, 4(2):447-459.
    • (2005) SIAM Multiscale Model. Simul. , vol.4 , Issue.2 , pp. 447-459
    • Abdulle, A.1
  • 2
    • 17644387561 scopus 로고    scopus 로고
    • Heterogeneous multiscale FEM for diffusion problem on rough surfaces
    • Abdulle A., Schwab C. Heterogeneous multiscale FEM for diffusion problem on rough surfaces. SIAM Multiscale Model. Simul. 2005, 3(1):195-220.
    • (2005) SIAM Multiscale Model. Simul. , vol.3 , Issue.1 , pp. 195-220
    • Abdulle, A.1    Schwab, C.2
  • 3
    • 33645990909 scopus 로고    scopus 로고
    • Analysis of a heterogeneous multiscale FEM for problems in elasticity
    • Abdulle A. Analysis of a heterogeneous multiscale FEM for problems in elasticity. Math. Models Methods Appl. Sci. 2006, 16(2):615-635.
    • (2006) Math. Models Methods Appl. Sci. , vol.16 , Issue.2 , pp. 615-635
    • Abdulle, A.1
  • 4
    • 79958699377 scopus 로고    scopus 로고
    • Heterogeneous multiscale methods with quadrilateral finite elements, in: Numerical Mathematics and Advanced Applications, Springer, Berlin
    • A. Abdulle, Heterogeneous multiscale methods with quadrilateral finite elements, in: Numerical Mathematics and Advanced Applications, Springer, Berlin, 2006, 743-751.
    • (2006) , pp. 743-751
    • Abdulle, A.1
  • 5
    • 55249110253 scopus 로고    scopus 로고
    • Finite element heterogeneous multiscale methods with near optimal computational complexity
    • Abdulle A., Engquist B. Finite element heterogeneous multiscale methods with near optimal computational complexity. SIAM Multiscale Model. Simul. 2007, 6(4):1059-1084.
    • (2007) SIAM Multiscale Model. Simul. , vol.6 , Issue.4 , pp. 1059-1084
    • Abdulle, A.1    Engquist, B.2
  • 6
    • 38749152055 scopus 로고    scopus 로고
    • Multiscale method based on discontinuous Galerkin methods for homogenization problems
    • Abdulle A. Multiscale method based on discontinuous Galerkin methods for homogenization problems. C.R. Acad. Sci. Paris, Ser. I 2008, 346(1-2):97-102.
    • (2008) C.R. Acad. Sci. Paris, Ser. I , vol.346 , Issue.1-2 , pp. 97-102
    • Abdulle, A.1
  • 7
    • 68049097915 scopus 로고    scopus 로고
    • The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs
    • Abdulle A. The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs. GAKUTO Int. Ser. Math. Sci. Appl. 2009, 31:135-184.
    • (2009) GAKUTO Int. Ser. Math. Sci. Appl. , vol.31 , pp. 135-184
    • Abdulle, A.1
  • 8
    • 68049087929 scopus 로고    scopus 로고
    • A short and versatile finite element multiscale code for homogenization problems
    • Abdulle A., Nonnenmacher A. A short and versatile finite element multiscale code for homogenization problems. Comput. Methods Appl. Mech. Engrg. 2009, 198(37-40):2839-2859.
    • (2009) Comput. Methods Appl. Mech. Engrg. , vol.198 , Issue.37-40 , pp. 2839-2859
    • Abdulle, A.1    Nonnenmacher, A.2
  • 9
    • 68949159804 scopus 로고    scopus 로고
    • A posteriori error analysis of the heterogeneous multiscale method for homogenization problems
    • Abdulle A., Nonnenmacher A. A posteriori error analysis of the heterogeneous multiscale method for homogenization problems. C.R. Acad. Sci. Paris, Ser. I 2009, 347(17-18):1081-1086.
    • (2009) C.R. Acad. Sci. Paris, Ser. I , vol.347 , Issue.17-18 , pp. 1081-1086
    • Abdulle, A.1    Nonnenmacher, A.2
  • 10
    • 79958697882 scopus 로고    scopus 로고
    • Discontinuous Galerkin finite element heterogeneous multiscale method for elliptic problems with multiple scales, preprint.
    • A. Abdulle, Discontinuous Galerkin finite element heterogeneous multiscale method for elliptic problems with multiple scales, preprint.
    • Abdulle, A.1
  • 13
    • 0000828256 scopus 로고
    • Generalized finite element methods: their performance and their relation to mixed methods
    • Babǔska I., Osborn E. Generalized finite element methods: their performance and their relation to mixed methods. SIAM J. Numer. Anal. 1983, 20:510-536.
    • (1983) SIAM J. Numer. Anal. , vol.20 , pp. 510-536
    • Babǔska, I.1    Osborn, E.2
  • 14
    • 0028483749 scopus 로고
    • Special finite element methods for a class of second order elliptic problems with rough coefficients
    • Babǔska I., Caloz G., Osborn E. Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J. Numer. Anal. 1994, 31:945-981.
    • (1994) SIAM J. Numer. Anal. , vol.31 , pp. 945-981
    • Babǔska, I.1    Caloz, G.2    Osborn, E.3
  • 16
    • 1542714643 scopus 로고    scopus 로고
    • Approximations of effective coefficients in stochastic homogenization
    • Bourgeat A., Piatnitski A. Approximations of effective coefficients in stochastic homogenization. Ann. Inst. H. Poincaré Prob. Stat. 2004, 40(2):153-165.
    • (2004) Ann. Inst. H. Poincaré Prob. Stat. , vol.40 , Issue.2 , pp. 153-165
    • Bourgeat, A.1    Piatnitski, A.2
  • 17
    • 79958716611 scopus 로고
    • Partial differential equations, in: Lectures in Applied Mathematics, Proceedings of the Summer Seminar, Boulder, CO
    • L. Bers, F. John, M. Schechter, Partial differential equations, in: Lectures in Applied Mathematics, Proceedings of the Summer Seminar, Boulder, CO, 1957.
    • (1957)
    • Bers, L.1    John, F.2    Schechter, M.3
  • 18
    • 12844279388 scopus 로고    scopus 로고
    • Discrétisations variationnelles de problèmes aux limites elliptiques
    • Springer-Verlag, Berlin
    • Bernardi C., Maday Y., Rapetti F. Discrétisations variationnelles de problèmes aux limites elliptiques. Mathématiques & Applications 2004, vol. 45. Springer-Verlag, Berlin.
    • (2004) Mathématiques & Applications , vol.45
    • Bernardi, C.1    Maday, Y.2    Rapetti, F.3
  • 19
    • 79958713836 scopus 로고    scopus 로고
    • AFEM@matlab: A Matlab Package of Adaptive Finite Element Methods, Technical Report
    • L. Chen, C.S. Zhang, AFEM@matlab: A Matlab Package of Adaptive Finite Element Methods, Technical Report, 2006.
    • (2006)
    • Chen, L.1    Zhang, C.S.2
  • 20
    • 0036906639 scopus 로고    scopus 로고
    • The finite element method for elliptic problems
    • SIAM, Philadelphia, PA
    • Ciarlet P. The finite element method for elliptic problems. Classics in Applied Mathematics 2002, vol. 40. SIAM, Philadelphia, PA.
    • (2002) Classics in Applied Mathematics , vol.40
    • Ciarlet, P.1
  • 23
    • 0000618557 scopus 로고
    • Sulla convergenza degli integrali dell'energia per operatori ellittici del secondo ordine
    • De Giorgi E., Spagnolo S. Sulla convergenza degli integrali dell'energia per operatori ellittici del secondo ordine. Boll. Un. Mat. Ital. 1973, 4(8):391-411.
    • (1973) Boll. Un. Mat. Ital. , vol.4 , Issue.8 , pp. 391-411
    • De Giorgi, E.1    Spagnolo, S.2
  • 24
    • 85128805678 scopus 로고    scopus 로고
    • The heterogeneous multiscale methods,
    • W. E, B. Engquist, The heterogeneous multiscale methods, Commun. Math. Sci. 1(1) (2003), 87-132.
    • (2003) Commun. Math. Sci. , vol.1 , Issue.1 , pp. 87-132
    • Engquist, W.E.B.1
  • 25
    • 13644268174 scopus 로고    scopus 로고
    • Analysis of the heterogeneous multiscale method for elliptic homogenization problems
    • E W., Ming P., Zhang P. Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J. Am. Math. Soc. 2005, 18(1):121-156.
    • (2005) J. Am. Math. Soc. , vol.18 , Issue.1 , pp. 121-156
    • E, W.1    Ming, P.2    Zhang, P.3
  • 26
    • 17644363709 scopus 로고    scopus 로고
    • High-dimensional finite elements for elliptic problems with multiple scales
    • Hoang V.H., Schwab C. High-dimensional finite elements for elliptic problems with multiple scales. SIAM Multiscale Model. Simul. 2005, 3(1):168-194.
    • (2005) SIAM Multiscale Model. Simul. , vol.3 , Issue.1 , pp. 168-194
    • Hoang, V.H.1    Schwab, C.2
  • 27
    • 0039982148 scopus 로고    scopus 로고
    • Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients
    • Hou T.Y., Wu X-H., Cai Z. Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comput. 1999, 68(227):913-943.
    • (1999) Math. Comput. , vol.68 , Issue.227 , pp. 913-943
    • Hou, T.Y.1    Wu, X.-H.2    Cai, Z.3
  • 29
  • 30
    • 0003218752 scopus 로고
    • The boundary value problems of mathematical physics
    • Springer-Verlag, New York Inc.
    • Ladyzhenskaya O.A. The boundary value problems of mathematical physics. Applied Mathematical Sciences 1985, vol. 49. Springer-Verlag, New York Inc.
    • (1985) Applied Mathematical Sciences , vol.49
    • Ladyzhenskaya, O.A.1
  • 31
  • 32
    • 0036090896 scopus 로고    scopus 로고
    • On the homogenization analysis of composite materials based on discretized fluctuations on the micro-structure
    • Miehe C., Schröder J., Bayreuther C. On the homogenization analysis of composite materials based on discretized fluctuations on the micro-structure. Acta Mech. 2002, 135:1-16.
    • (2002) Acta Mech. , vol.135 , pp. 1-16
    • Miehe, C.1    Schröder, J.2    Bayreuther, C.3
  • 33
    • 0036959161 scopus 로고    scopus 로고
    • Convergence of adaptive finite element methods
    • Morin P., Nochetto R.H., Siebert K.G. Convergence of adaptive finite element methods. SIAM Rev. 2002, 44(4):631-658.
    • (2002) SIAM Rev. , vol.44 , Issue.4 , pp. 631-658
    • Morin, P.1    Nochetto, R.H.2    Siebert, K.G.3
  • 34
    • 0002615068 scopus 로고    scopus 로고
    • H-convergence
    • Birkhäuser Boston, Boston, MA, 21-43, Topics in the Mathematical Modelling of Composite Materials
    • Murat F., Tartar L. H-convergence. Progr. Nonlinear Differential Equations Appl. 1997, vol. 31. Birkhäuser Boston, Boston, MA, 21-43.
    • (1997) Progr. Nonlinear Differential Equations Appl. , vol.31
    • Murat, F.1    Tartar, L.2
  • 35
    • 0034950892 scopus 로고    scopus 로고
    • Homogenization and multigrid
    • Neuss N., Jäger W., Wittum G. Homogenization and multigrid. Computing 2001, 66(1):1-26.
    • (2001) Computing , vol.66 , Issue.1 , pp. 1-26
    • Neuss, N.1    Jäger, W.2    Wittum, G.3
  • 36
    • 0000315621 scopus 로고
    • A general convergence result for a functional related to the theory of homogenization
    • Nguetseng G. A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 1989, 20(3):608-623.
    • (1989) SIAM J. Math. Anal. , vol.20 , Issue.3 , pp. 608-623
    • Nguetseng, G.1
  • 37
    • 79958753559 scopus 로고    scopus 로고
    • Adaptive finite element methods for elliptic PDEs, Lecture Notes,
    • R.H. Nochetto, Adaptive finite element methods for elliptic PDEs, Lecture Notes, http://www-users.math.umd.edu/rhn/lectures.html.
    • Nochetto, R.H.1
  • 38
    • 34249311837 scopus 로고    scopus 로고
    • Multiscale modeling of physical phenomena: adaptive control of models
    • Oden J.T., Prudhomme S., Romkes A., Bauman P. Multiscale modeling of physical phenomena: adaptive control of models. SIAM J. Sci. Comput. 2006, 28(6):2359-2389.
    • (2006) SIAM J. Sci. Comput. , vol.28 , Issue.6 , pp. 2359-2389
    • Oden, J.T.1    Prudhomme, S.2    Romkes, A.3    Bauman, P.4
  • 39
    • 33644783757 scopus 로고    scopus 로고
    • A posteriori error estimates for the heterogeneous multiscale finite element method for elliptic homogenization problems
    • Ohlberger M. A posteriori error estimates for the heterogeneous multiscale finite element method for elliptic homogenization problems. Multiscale Model. Simul. 2005, 4(1):88-114.
    • (2005) Multiscale Model. Simul. , vol.4 , Issue.1 , pp. 88-114
    • Ohlberger, M.1
  • 40
    • 27144549033 scopus 로고    scopus 로고
    • Design of adaptive finite element software: the finite element toolbox ALBERTA
    • Springer-Verlag, Berlin
    • Schmidt A., Siebert K.G. Design of adaptive finite element software: the finite element toolbox ALBERTA. Lecture Notes in Computational Science and Engineering 2005, vol. 42. Springer-Verlag, Berlin.
    • (2005) Lecture Notes in Computational Science and Engineering , vol.42
    • Schmidt, A.1    Siebert, K.G.2
  • 41
    • 0035919604 scopus 로고    scopus 로고
    • A class of general algorithms for multi-scale analyses of heterogeneous media
    • Terada K., Kikuchi N. A class of general algorithms for multi-scale analyses of heterogeneous media. Comput. Methods Appl. Mech. Engrg. 2001, 190(40-41):5427-5464.
    • (2001) Comput. Methods Appl. Mech. Engrg. , vol.190 , Issue.40-41 , pp. 5427-5464
    • Terada, K.1    Kikuchi, N.2
  • 43
    • 0032836187 scopus 로고    scopus 로고
    • Accurate scale up of two phase flow using renormalization and nonuniform coarsening
    • Wallstrom T.C., Hou S., Christie M.A., Durlofsky L.J., Sharp D.H. Accurate scale up of two phase flow using renormalization and nonuniform coarsening. Comput. Geosci. 1999, 3(1):67-87.
    • (1999) Comput. Geosci. , vol.3 , Issue.1 , pp. 67-87
    • Wallstrom, T.C.1    Hou, S.2    Christie, M.A.3    Durlofsky, L.J.4    Sharp, D.H.5
  • 44
    • 0036900841 scopus 로고    scopus 로고
    • Multiscale asymptotic homogenization for multiphysics problems with multiple spatial and temporal scales: a coupled thermo-viscoelastic example problem
    • Yu Q., Fish J. Multiscale asymptotic homogenization for multiphysics problems with multiple spatial and temporal scales: a coupled thermo-viscoelastic example problem. Int. J. Solids Struct. 2002, 39(26):6429-6452.
    • (2002) Int. J. Solids Struct. , vol.39 , Issue.26 , pp. 6429-6452
    • Yu, Q.1    Fish, J.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.