메뉴 건너뛰기




Volumn 4, Issue 1, 2005, Pages 36-87

Finite difference approximation of homogenization problems for elliptic equations

Author keywords

Bloch waves; Finite differences; Homogenization; Multiscale

Indexed keywords

APPROXIMATION THEORY; BOUNDARY CONDITIONS; EIGENVALUES AND EIGENFUNCTIONS; ESTIMATION; FOURIER TRANSFORMS; PARTIAL DIFFERENTIAL EQUATIONS;

EID: 32544445185     PISSN: 15403459     EISSN: 15403467     Source Type: Journal    
DOI: 10.1137/040606314     Document Type: Article
Times cited : (8)

References (34)
  • 1
    • 0000741301 scopus 로고
    • Finite difference approximations for partial differential equations with rapidly oscillating coefficients
    • M. Avellaneda, Th. Y. Hou, and G. Papanicolaou, Finite difference approximations for partial differential equations with rapidly oscillating coefficients, RAIRO Modél. Math. Anal. Numér., 25 (1991), pp. 693-710.
    • (1991) RAIRO Modél. Math. Anal. Numér. , vol.25 , pp. 693-710
    • Avellaneda, M.1    Hou, Th.Y.2    Papanicolaou, G.3
  • 3
    • 34250940025 scopus 로고
    • Über die quantenmechanik der electronen in kristallgittern
    • F. Bloch, Über die quantenmechanik der electronen in kristallgittern, Z. Phys., 52 (1928), pp. 555-600.
    • (1928) Z. Phys. , vol.52 , pp. 555-600
    • Bloch, F.1
  • 4
    • 84968498740 scopus 로고
    • Convergence estimates for essentially positive type discrete Dirichlet problems
    • J. H. Bramble, B. E. Hubbarb, and V. Thomée, Convergence estimates for essentially positive type discrete Dirichlet problems, Math. Comp., 23 (1969), pp. 695-709.
    • (1969) Math. Comp. , vol.23 , pp. 695-709
    • Bramble, J.H.1    Hubbarb, B.E.2    Thomée, V.3
  • 5
    • 33644680704 scopus 로고
    • Propriétés d'approximation dans les espaces de Sobolev de systèmes de polynômes orthogonaux
    • C. Canuto and A. Quarteroni, Propriétés d'approximation dans les espaces de Sobolev de systèmes de polynômes orthogonaux, C. R. Acad. Sci. Paris Sér. A-B, 290 (1980), pp. 925-928.
    • (1980) C. R. Acad. Sci. Paris Sér. A-B , vol.290 , pp. 925-928
    • Canuto, C.1    Quarteroni, A.2
  • 6
    • 0037414944 scopus 로고    scopus 로고
    • Numerical methods for elliptic partial differential equations with rapidly oscillating coefficients
    • C. Conca and S. Natesan, Numerical methods for elliptic partial differential equations with rapidly oscillating coefficients, Comput. Methods Appl. Mech. Engrg., 192 (2003), pp. 47-76.
    • (2003) Comput. Methods Appl. Mech. Engrg. , vol.192 , pp. 47-76
    • Conca, C.1    Natesan, S.2
  • 7
    • 0036375626 scopus 로고    scopus 로고
    • Bloch approximation in homogenization and applications
    • C. Conca, R. Orive, and M. Vanninathan, Bloch approximation in homogenization and applications, SIAM J. Math. Anal., 33 (2002), pp. 1166-1198.
    • (2002) SIAM J. Math. Anal. , vol.33 , pp. 1166-1198
    • Conca, C.1    Orive, R.2    Vanninathan, M.3
  • 8
    • 23744499919 scopus 로고    scopus 로고
    • Bloch approximation in homogenization on bounded domains
    • C. Conca, R. Orive, and M. Vanninathan, Bloch approximation in homogenization on bounded domains, Asymptot. Anal., 41 (2005), pp. 71-91.
    • (2005) Asymptot. Anal. , vol.41 , pp. 71-91
    • Conca, C.1    Orive, R.2    Vanninathan, M.3
  • 9
    • 0003359106 scopus 로고
    • Fluids and Periodic Structures
    • John Wiley & Sons, Chichester, Masson, Paris
    • C. Conca, J. Planchard, and M. Vanninathan, Fluids and Periodic Structures, RAM: Res. Appl. Math. 38, John Wiley & Sons, Chichester, Masson, Paris, 1995.
    • (1995) RAM: Res. Appl. Math. , vol.38
    • Conca, C.1    Planchard, J.2    Vanninathan, M.3
  • 10
    • 0031352502 scopus 로고    scopus 로고
    • Homogenization of periodic structures via Bloch decomposition
    • C. Conca and M. Vanninathan, Homogenization of periodic structures via Bloch decomposition, SIAM J. Appl. Math., 57 (1997), pp. 1639-1659.
    • (1997) SIAM J. Appl. Math. , vol.57 , pp. 1639-1659
    • Conca, C.1    Vanninathan, M.2
  • 12
    • 0000713961 scopus 로고    scopus 로고
    • Convergence of a nonconforming multiscale finite element method
    • Y. R. Efendiev, Th. Y. Hou, and X.-H. Wu, Convergence of a nonconforming multiscale finite element method, SIAM J. Numer. Anal., 37 (2000), pp. 888-910.
    • (2000) SIAM J. Numer. Anal. , vol.37 , pp. 888-910
    • Efendiev, Y.R.1    Hou, Th.Y.2    Wu, X.-H.3
  • 13
    • 21244447771 scopus 로고    scopus 로고
    • Numerical homogenization of nonlinear random parabolic operators
    • Y. R. Efendiev and A. Pankov, Numerical homogenization of nonlinear random parabolic operators, Multiscale Model. Simul., 2 (2004), pp. 237-268.
    • (2004) Multiscale Model. Simul. , vol.2 , pp. 237-268
    • Efendiev, Y.R.1    Pankov, A.2
  • 14
    • 0036002578 scopus 로고    scopus 로고
    • Multiscale finite element for problems with highly oscillatory coefficients
    • Y. R. Efendiev and X.-H. Wu, Multiscale finite element for problems with highly oscillatory coefficients, Numer. Math., 90 (2002), pp. 459-486.
    • (2002) Numer. Math. , vol.90 , pp. 459-486
    • Efendiev, Y.R.1    Wu, X.-H.2
  • 15
    • 0141599961 scopus 로고
    • Computation of oscillatory solutions to hyperbolic differential equations
    • Lecture Notes in Math. 1270, Springer-Verlag, Berlin
    • B. Engquist, Computation of oscillatory solutions to hyperbolic differential equations, in Nonlinear Hyperbolic Problems, Lecture Notes in Math. 1270, Springer-Verlag, Berlin, 1987, pp. 10-22.
    • (1987) Nonlinear Hyperbolic Problems , pp. 10-22
    • Engquist, B.1
  • 16
    • 0010820783 scopus 로고
    • Particle method approximation of oscillatory solutions to hyperbolic differential equations
    • B. Engquist and Th. Y. Hou, Particle method approximation of oscillatory solutions to hyperbolic differential equations, SIAM J. Numer. Anal., 26 (1989), pp. 289-319.
    • (1989) SIAM J. Numer. Anal. , vol.26 , pp. 289-319
    • Engquist, B.1    Hou, Th.Y.2
  • 17
    • 84990617897 scopus 로고
    • Numerical methods for oscillatory solutions to hyperbolic problems
    • B. Engquist and J. G. Liu, Numerical methods for oscillatory solutions to hyperbolic problems, Comm. Pure Appl. Math., 46 (1993), pp. 1327-1361.
    • (1993) Comm. Pure Appl. Math. , vol.46 , pp. 1327-1361
    • Engquist, B.1    Liu, J.G.2
  • 18
    • 0001739202 scopus 로고    scopus 로고
    • Convergence of a multigrid method for elliptic equations with highly oscillatory coefficients
    • B. Engquist and E. Luo, Convergence of a multigrid method for elliptic equations with highly oscillatory coefficients, SIAM J. Numer. Anal., 34 (1997), pp. 2254-2273.
    • (1997) SIAM J. Numer. Anal. , vol.34 , pp. 2254-2273
    • Engquist, B.1    Luo, E.2
  • 20
    • 0031161210 scopus 로고    scopus 로고
    • A multiscale finite element method for elliptic problems in composite materials and porous media
    • Th. Y. Hou and X.-H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134 (1997), pp. 169-189.
    • (1997) J. Comput. Phys. , vol.134 , pp. 169-189
    • Hou, Th.Y.1    Wu, X.-H.2
  • 21
    • 0039982148 scopus 로고    scopus 로고
    • Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients
    • Th. Y. Hou, X.-H. Wu, and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp., 68 (1999), pp. 913-943.
    • (1999) Math. Comp. , vol.68 , pp. 913-943
    • Hou, Th.Y.1    Wu, X.-H.2    Cai, Z.3
  • 23
    • 0002615845 scopus 로고
    • Averaging of difference schemes
    • S. M. Kozlov, Averaging of difference schemes, Math. USSR-Sb., 57 (1987), 351-369.
    • (1987) Math. USSR-Sb. , vol.57 , pp. 351-369
    • Kozlov, S.M.1
  • 25
    • 0040788429 scopus 로고    scopus 로고
    • Generalized p-FEM in homogenization
    • A.-M. Matache, I. Babuska, and C. Schwab, Generalized p-FEM in homogenization, Numer. Math., 86 (2000), pp. 319-375.
    • (2000) Numer. Math. , vol.86 , pp. 319-375
    • Matache, A.-M.1    Babuska, I.2    Schwab, C.3
  • 28
    • 0000997571 scopus 로고
    • Quasi-Monte Carlo methods and pseudo-random numbers
    • H. Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc., 84 (1978), pp. 957-1041.
    • (1978) Bull. Amer. Math. Soc. , vol.84 , pp. 957-1041
    • Niederreiter, H.1
  • 29
    • 0348065096 scopus 로고
    • Error estimates for the numerical solution of elliptic differential equations
    • J. Nitsche and J. C. C. Nitsche, Error estimates for the numerical solution of elliptic differential equations, Arch. Rational Mech. Anal., 5 (1960), pp. 293-306.
    • (1960) Arch. Rational Mech. Anal. , vol.5 , pp. 293-306
    • Nitsche, J.1    Nitsche, J.C.C.2
  • 30
    • 0001170610 scopus 로고
    • Partial differential equations with periodic coefficients and Bloch waves in crystals
    • F. Odeh and J. Keller, Partial differential equations with periodic coefficients and Bloch waves in crystals, J. Math. Phys., 5 (1964), pp. 1499-1504.
    • (1964) J. Math. Phys. , vol.5 , pp. 1499-1504
    • Odeh, F.1    Keller, J.2
  • 31
    • 0035527614 scopus 로고    scopus 로고
    • Homogenization of elliptic difference operators
    • A. Piatnitski and E. Remy, Homogenization of elliptic difference operators, SIAM J. Math. Anal., 33 (2001), pp. 53-83.
    • (2001) SIAM J. Math. Anal. , vol.33 , pp. 53-83
    • Piatnitski, A.1    Remy, E.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.