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A LOCAL CONSERVATIVE MULTISCALE METHOD FOR ELLIPTIC PROBLEMS WITH OSCILLATING COEFFICIENTS

  • JEON, YOUNGMOK (DEPARTMENT OF MATHEMATICS, AJOU UNIVERSITY) ;
  • PARK, EUN-JAE (DEPARTMENT OF COMPUTATIONAL SCIENCE AND ENGINEERING, YONSEI UNIVERSITY)
  • Received : 2020.06.05
  • Accepted : 2020.06.16
  • Published : 2020.06.25

Abstract

A new multiscale finite element method for elliptic problems with highly oscillating coefficients are introduced. A hybridization yields a locally flux-conserving numerical scheme for multiscale problems. Our approach naturally induces a homogenized equation which facilitates error analysis. Complete convergence analysis is given and numerical examples are presented to validate our analysis.

References

  1. T. Arbogast, Homogenization-based mixed multiscale finite elements for problems with anisotropy, Multiscale Modeling and Simulation 9 (2) (2011), pp. 624-653.
  2. A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic Analysis for periodic Structures, North-Holland, Amsterdam, 1978.
  3. T. J. R. Hughes, G. R. Feijoo, L. Mazzei, J.-B. Quincy, The variational multiscale method. A paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg. 166 (1998) 3-24.
  4. J. T. Oden, K. S. Vemaganti, Adaptive hierarchical modeling of heterogeneous structures, Phys. D 133 (1999) 404-415.
  5. Z. Chen, T. Y. Hou, A mixed multiscale finite element method for elliptic problems with oscillating coefficients, Math. Comp. 72 (2003) 541-576.
  6. Y. Efendiev, T. Hou, Multiscale Finite Element Methods, Theory and Applications, Surveys and Tutorials in the Applied Mathematical Sciences, vol.4. Springer, New York, 2009.
  7. T. Y. Hou, X.-H.Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys. 134 (1997) 169-189.
  8. T. Y. Hou, X.-H. Wu, Y. Zhang, Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation, Commun. Math. Sci. 2 (2) (2004) 185-205.
  9. F. Brezzi, Interacting with the subgrid world, in: Numerical Analysis 1999 (Dundee), in: Chapman & Hall/CRC Res. Notes Math., vol. 420, Chapman & Hall/CRC, Boca Raton, FL, 2000, pp. 69-82.
  10. G. Sangalli, Capturing small scales in elliptic problems using a residual-free bubbles finite element method, Multiscale Model. Simul. 1 (2003) 485-503.
  11. W. E, B. Engquist, The heterogeneous multi-scale methods, Commun. Math. Sci. 1 (1) (2003) 87-132.
  12. R. Du, P. Ming, Convergence of the heterogeneous multiscale finite element method for ellitpic problems with nonsmoothe microstructures, Multisclae Model. Simul. 8(5) (2010) 1770-1783.
  13. C. Schwab, A.-M. Matache, Generalized FEM for homogenization problems, in: Multiscale and Multiresolution Methods, in: Lect. Notes Comput. Sci. Eng., vol. 20, Springer-Verlag, Berlin, 2002, pp. 197-237.
  14. T. Arbogast, G. Pencheva, M. F. Wheeler, and I. Yotov, A multiscale mortar mixed finite element method. Multiscale Model. Simul. Vol. 6, No. 1, (2007), pp. 319-346.
  15. M.-Y. KIM, E.-J. PARK, S. G. THOMAS, AND M. F. WHEELER, A multiscale mortar mixed finite element method for slightly compressible flows in porous media, J. Korean Math. Soc. 44 (2007), No. 5, pp. 1103-1119
  16. M. Arshad, E.-J. Park, and D.-w. Shin, Analysis of Multiscale Mortar Mixed Approximation of Nonlinear Elliptic Equations, Computers & Mathematics with Applications, Vol 75, No. 2, (2018), 401-418
  17. M.-Y. KIM AND M. F. WHEELER, A multiscale discontinuous galerkin method for convection-diffusion-reaction problems, Computers & Mathematics with Applications, 68 (2014), pp. 2251-2261.
  18. Y. Jeon, A multiscale cell boundary element method for elliptic problems, Appl. Numer. Math. 59 (11) (2009) 2801-2813.
  19. Y. Jeon, E.-J. Park, Nonconforming cell boundary element methods for elliptic problems on triangular mesh, Appl. Numer. Math. 58 (6) (2008) 800-814.
  20. Y. Jeon, E.-J. Park, A hybrid discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal. 48 (5) (2010) 1968-1983.
  21. Y. Jeon, E.-J. Park, New locally conservative finite element methods on a rectangular mesh, Numerische Mathematik, 123 (2013), no.1, 97-119.
  22. Y. Jeon, and E.-J. Park, D.-w. Shin, Hybrid Spectral Difference Methods for an Elliptic Equation, Comput. Methods Appl. Math. vol.17 no.2 (2017), 253-267.
  23. M.-Y. KIM AND M. F. WHEELER, Coupling discontinuous Galerkin discretizations using mortar finite elements for advection-diffusion-reaction problems, Computers & Mathematics with Applications, 67 (2014), pp. 181-198.
  24. Y. Jeon, E.-J. Park, D. Sheen, A hybridized finite element method for the Stokes problem, Computers & Mathematics with Applications Vol. 68, No. 12 Part B, (2014), 2222-2232.
  25. D.-w. Shin, Y. Jeon, and E.-J. Park, A hybrid discontinuous Galerkin method for advection-diffusion-reaction problems, Applied Numerical Mathematics, 95 (2015), 292-303.
  26. L. Zhao and E.-J. Park. A new hybrid staggered discontinuous Galerkin method on general meshes. Journal of Scientific Computing, 82:12, 2020.
  27. P. Jenny, S. H. Lee, H. A. Tchelepi, Multi-scale finite-volume method for elliptic problems in subsurface flow simulation, J. Comput. Phys. 187 (2003) 47-67.
  28. V. V. Jikov, S. M. Kozlov, O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, Heidelberg, 1994.
  29. G. Allaire, R. Brizzi, A multiscale finite element method for numerical homogenization, Multiscale Model. Simul. 4 (3) (2005) 790-812.
  30. A. Abdulle, W. E, Finite difference HMM for homogenization problems, J. Comput. Phys. 191 (2003) 18-39.
  31. M. G. Larson and A.Malqvist, Adaptive variational multiscale methods based on a posteriori error estimation: Energy norm estimates for elliptic problems, Computer Methods in Applied Mechanics and Engineering 196 (2007), no. 21-24, 2313-2324.