Matrix-Dependent Multigrid Homogenization for Diffusion Problems

For problems with strongly varying or discontinuous diffusion coefficients we present a method to compute coarse-scale operators and to approximately determine the effective diffusion tensor on the coarse-scale level. The approach is based on techniques that are used in multigrid, such as matrix-dependent prolongations and the construction of coarse-grid operators by means of the Galerkin approximation. In numerical experiments we compare our multigrid-homogenization method with continuous homogenization, renormalization, and simple averaging approaches.

[1]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .

[2]  J. Bourgat.  Numerical experiments of the homogenization method , 1979 .

[3]  S. Kesavan.  Homogenization of elliptic eigenvalue problems: Part 1 , 1979 .

[4]  S. Kozlov.  AVERAGING OF RANDOM OPERATORS , 1980 .

[5]  E. Sanchez-Palencia.  Non-Homogeneous Media and Vibration Theory , 1980 .

[6]  A. Brandt,et al.  The Multi-Grid Method for the Diffusion Equation with Strongly Discontinuous Coefficients , 1981 .

[7]  J. Dendy.  Black box multigrid , 1982 .

[8]  J. E. Dendy.  Black box multigrid for nonsymmetric problems , 1983 .

[9]  G. Dagan.  Statistical Theory of Groundwater Flow and Transport: Pore to Laboratory, Laboratory to Formation, and Formation to Regional Scale , 1986 .

[10]  J. Koplik,et al.  Multiphase Flow in Porous Media: Mechanics, Mathematics, and Numerics , 1988 .

[11]  P. M. De Zeeuw,et al.  Matrix-dependent prolongations and restrictions in a blackbox multigrid solver , 1990 .

[12]  J. E. Dendy.  Black box multigrid for periodic and singular problems , 1988 .

[13]  N. Bakhvalov,et al.  Homogenisation: Averaging Processes in Periodic Media , 1989 .

[14]  N. Bakhvalov,et al.  Homogenisation: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of Composite Materials , 1989 .

[15]  P. King.  The use of renormalization for calculating effective permeability , 1989 .

[16]  P. Vassilevski,et al.  Algebraic multilevel preconditioning methods. I , 1989 .

[17]  O. Axelsson,et al.  Algebraic multilevel preconditioning methods, II , 1990 .

[18]  B. Amaziane,et al.  Numerical Simulation and Homogenization of Two-Phase Flow in Heterogeneous Porous Media , 1991 .

[19]  R. Showalter,et al.  Micro-Structure Models of Diffusion in Fissured Media* , 1991 .

[20]  G. D. Maso,et al.  An Introduction to-convergence , 1993 .

[21]  B. Engquist,et al.  Multigrid methods for differential equations with highly oscillatory coefficients , 1993 .

[22]  Michael Vogelius,et al.  First-Order Corrections to the Homogenized Eigenvalues of a Periodic Composite Medium , 1993, SIAM J. Appl. Math..

[23]  V. Zhikov,et al.  Homogenization of Differential Operators and Integral Functionals , 1994 .

[24]  A. Reusken,et al.  Multigrid with Matrix-dependent Transfer Operators for Convection-diffusion Problems , 1994 .

[25]  M. Edwards,et al.  Multigrid and Renormalization for Reservoir Simulation , 1994 .

[26]  Michael Andrew Christie,et al.  Upscaling for reservoir simulation , 1996 .

[27]  Robert E. O'Malley,et al.  Introduction to Perturbation Methods (M. H. Holmes) , 1996, SIAM Rev..