On Wavelet-Based Numerical Homogenization

Recently, a wavelet-based method was introduced for the systematic derivation of subgrid scale models in the numerical solution of partial differential equations. Starting from a discretization of the multiscale differential operator, the discrete operator is represented in a wavelet space and projected onto a coarser subspace. The coarse (homogenized) operator is then replaced by a sparse approximation to increase the efficiency of the resulting algorithm.In this work we show how to improve the efficiency of this numerical homogenization method by choosing a different compact representation of the homogenized operator. In two dimensions our approach for obtaining a sparse representation is significantly simpler than the alternative sparse representations. $L^{\infty}$ error estimates are derived for a sample elliptic problem. An additional improvement we propose is a natural fine-scales correction that can be implemented in the final homogenization step. This modification of the scheme improves the resol...

[1]  Gabriel Wittum,et al.  Filtering Decompositions with Respect to Adaptive Test Vectors , 1998 .

[2]  B. Engquist,et al.  A Contribution to Wavelet-Based Subgrid Modeling☆☆☆ , 1999 .

[3]  Tony F. Chan,et al.  The Interface Probing Technique in Domain Decomposition , 1992, SIAM J. Matrix Anal. Appl..

[4]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[5]  Gregory Beylkin,et al.  On the Adaptive Numerical Solution of Nonlinear Partial Differential Equations in Wavelet Bases , 1997 .

[6]  Y. Meyer.  Wavelets and Operators , 1993 .

[7]  Mihai Dorobantu,et al.  Walvelet-based algorithms for fast PDE solvers , 1995 .

[8]  T. Hou,et al.  Particle method approximation of oscillatory solutions to hyperbolic differential equations , 1989 .

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

[10]  O. Axelsson.  Iterative solution methods , 1995 .

[11]  B. Engquist,et al.  Wavelet-Based Numerical Homogenization with Applications , 2002 .

[12]  Thomas Y. Hou,et al.  A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media , 1997 .

[13]  Wolfgang Hackbusch.  The frequency decomposition multi-grid method , 1989 .

[14]  Christoph Schwab,et al.  Finite Dimensional Approximations for Elliptic Problems with Rapidly Oscillating Coefficients , 2002 .

[15]  A. Gilbert.  A Comparison of Multiresolution and Classical One-Dimensional Homogenization Schemes , 1998 .

[16]  G. Beylkin.  On Multiresolution Methods in Numerical Analysis , 2007 .

[17]  G. Beylkin.  Wavelets and Fast Numerical Algorithms , 1993, comp-gas/9304004.

[18]  B. Engquist,et al.  Wavelet-Based Numerical Homogenization , 1998 .

[19]  C. Schwab,et al.  Generalized FEM for Homogenization Problems , 2002 .

[20]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[21]  Wolfgang Hackbusch,et al.  The frequency decomposition multi-grid method , 1992 .

[22]  P. Laguna,et al.  Signal Processing , 2002, Yearbook of Medical Informatics.

[23]  G. Beylkin,et al.  A Multiresolution Strategy for Numerical Homogenization , 1995 .

[24]  Thomas Y. Hou,et al.  Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients , 1999, Math. Comput..

[25]  Stephan Knapek,et al.  Matrix-Dependent Multigrid Homogenization for Diffusion Problems , 1998, SIAM J. Sci. Comput..

[26]  G. Beylkin,et al.  A Multiresolution Strategy for Reduction of Elliptic PDEs and Eigenvalue Problems , 1998 .