Corrector theory for msfem and Hmm in random media

Multiscale Modeling and Simulation

Volumn 9, Issue 4, 2011, Pages 1549-1587

  Bal, Guillaume ^a   Jing, Wenjia ^b  

^a Department of Earth and Environmental Sciences   (United States)

^b ECOLE NORMALE SUPÉRIEURE   (France)

Author keywords

Corrector test; Correlation ranges; Equations with random coefficients; Heterogeneous multiscale method; Multiscale finite element method

Indexed keywords

CHOICE OF PARAMETERS; COMPUTATIONAL COSTS; CONTINUOUS MODELS; CORRECTOR TEST; CORRELATION LENGTHS; ELLIPTIC EQUATIONS; ERGODICS; FINITE ELEMENT; FLUCTUATION THEORY; GAUSSIAN PROCESSES; HETEROGENEOUS MULTISCALE METHOD; LONG RANGE CORRELATIONS; MESH SIZE; MODIFIED SCHEME; MULTISCALE ALGORITHMS; MULTISCALE FINITE ELEMENT METHOD; RANDOM COEFFICIENTS; RANDOM ENVIRONMENT; RANDOM FLUCTUATION; RANDOM MEDIA; RANDOM MEDIUM;

FINITE ELEMENT METHOD; GAUSSIAN DISTRIBUTION; GAUSSIAN NOISE (ELECTRONIC); PARTIAL DIFFERENTIAL EQUATIONS;

ALGORITHMS;

EID: 84856550296     PISSN: 15403459     EISSN: 15403467     Source Type: Journal    
DOI: 10.1137/100815918     Document Type: Article

References (25)

1. T. ARBOGAST, Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems, SIAM J. Numer. Anal., 42 (2004), pp. 576-598. (Pubitemid 40645206)
2. G. BAL, Central limits and homogenization in random media, Multiscale Model. Simul., 7 (2008), pp. 677-702.
3. G. BAL, J. GARNIER, S. MOTSCH, AND V. PERRIER, Random integrals and correctors in homogenization, Asymptot. Anal., 59 (2008), pp. 1-26.
4. G. BAL, R. GHANEM, AND I. LANGMORE, Large deviation theory for a homogenized and corrected elliptic ODE, J. Differential Equations, 251 (2011), pp. 1864-1902.
5. P. BILLINGSLEY, Convergence of Probability Measures, Wiley Series in Probability and Statistics: Probability and Statistics, 2nd ed., John Wiley & Sons, New York, 1999.
6. A. BOURGEAT AND A. PIATNITSKI, Estimates in probability of the residual between the random and the homogenized solutions of one-dimensional second-order operator, Asymptot. Anal., 21 (1999), pp. 303-315. (Pubitemid 129565616)
7. Z. CHEN AND T. Y. SAVCHUK, Analysis of the multiscale finite element method for nonlinear and random homogenization problems, SIAM J. Numer. Anal., 46 (2008), pp. 260-279.
8. P. DOUKHAN, Mixing, Lecture Notes in Statistics 85, Springer-Verlag, New York, 1994.
9. B. ENGQUIST, X. LI, W. REN, AND E. VANDEN-EIJNDEN, Heterogeneous multiscale methods: A review, Commun. Comput. Phys., 2 (2007), pp. 367-450.
10. P.MING, AND P. ZHANG, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc., 18 (2005), pp. 121-156. (Pubitemid 40230338)
11. Y. EFENDIEV, The Multiscale Finite Element Method (MsFEM) and Its Applications, Ph.D. thesis, California Institute of Technology, Pasadena, CA, 1999.
12. Y. R. EFENDIEV, T. Y. HOU, AND X.-H. WU, Convergence of a nonconforming multiscale finite element method, SIAM J. Numer. Anal., 37 (2000), pp. 888-910.
13. L. C. EVANS, Partial Differential Equations, Graduate Studies in Mathematics 19, AMS, Providence, RI, 1998.
14. R. FIGARI, E. ORLANDI, AND G. PAPANICOLAOU, Mean field and Gaussian approximation for partial differential equations with random coefficients, SIAM J. Appl. Math., 42 (1982), pp. 1069-1077. (Pubitemid 13461311)
15. T. Y. HOU, Numerical approximations to multiscale solutions in partial differential equations, in Frontiers in Numerical Analysis (Durham 2002), Universitext, Springer-Verlag, Berlin, 2003, pp. 241-301.
16. T. 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. (Pubitemid 127175548)
17. T. 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.
18. V. V. JIKOV, S. M. KOZLOV, AND O. A. OLEINIK, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, New York, 1994.
19. I. KARATZAS AND S. E. SHREVE, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics 113, 2nd ed., Springer-Verlag, New York, 1991.
20. S. M. KOZLOV, The averaging of random operators, Mat. Sb., 109 (1979), pp. 188-202, 327.
21. E. H. LIEB AND M. LOSS, Analysis, Graduate Studies in Mathematics 14, 2nd ed., AMS, Providence, RI, 2001.
22. G. C. PAPANICOLAOU AND S. R. S. VARADHAN, Boundary value problems with rapidly oscillating random coefficients, in Random Fields, Vol. I, II (Esztergom, 1979), Colloq. Math. Soc. János Bolyai 27, North-Holland, Amsterdam, 1981, pp. 835-873.
23. V. PIPIRAS AND M. S. TAQQU, Integration questions related to fractional Brownian motion, Probab. Theory Related Fields, 118 (2000), pp. 251-291. (Pubitemid 33228653)
24. A. QUARTERONI AND A. VALLI, Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics 23, Springer-Verlag, Berlin, 1994.
25. M. REED AND B. SIMON, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975.