×

Explicit formulas for wavelet-homogenized coefficients of elliptic operators. (English) Zbl 1121.65133

The author employs the method of M. E. Brewster and G. Beylkin [Appl. Comput. Harmon. Anal. 2, No. 4, 327–349 (1995; Zbl 0840.65047)] to obtain wavelet homogenized equations for discrete approximations to equations involving elliptic differential operators (i) \(d/dx\mu(x)d/dx\), (ii) \(\mu(x)\Delta\). Given a vector \(u(k)\), \(k= 0,\dots, N-1\) expressed on an orthogonal wavelet basis, filters are employed to divide \(u\) into coarse and fine components \(u= u_s+ u_d\), \(u_s(k)\), \(u_d(k)\), \(k= 0,\dots, N/2-1\). An \(N\times N\) matrix \(P\) acting on \(u\) goes into block form \(Pu= \left[\begin{smallmatrix} A_p & B_p\\ C_p & T_p\end{smallmatrix}\right] u\) and (iii) \(P=f\) becomes \(A_p u_d+ B_p u_s= f_d\), \(C_p u_d+ T_p u_s= f_s\). Eliminating \(u_d\) leads to an equation in the \(u_s\) vectors (iv) \(R_p u_s= f_s- C_p A^{-1}_p f_d\), \(R_p= T_p- C_p A^{-1}_p B_p\). The solution of (iv) is the coarse scale average of the solution of (iii) and is a homogenization of (iii).
Let (v) \(S=\Delta_-\,V\,\vee\Delta_+\), (vi) \(M= VL\) be discrete approximations to (i), (ii), where \(V\) is a diagonal matrix with entries \(\mu(k+{1\over 2}/N)\). If the values of \(\mu\) are periodic then the Fourier symbols of \(R_S\), \(R_M\) are detemined. The \(R_S\) symbol is found explicitly when \(\mu\) has alternating values \(\alpha\), \(\beta\). When \(\mu\) is not periodic \(R_S\) and \(R_M\) are approximated two ways using a mass-lumping procedure. Results of numerical tests of the accuracy of the approximations are described and show that the methods are effective.

MSC:

65T60 Numerical methods for wavelets
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
47F05 General theory of partial differential operators

Citations:

Zbl 0840.65047

Software:

JDQR; JDQZ
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alpert, B., A class of bases in \(L^2\) for the sparse representation of integral operators, SIAM J. Math. Anal., 24, 1, 246-262 (1993) · Zbl 0764.42017
[2] Andersson, U.; Engquist, B.; Ledfelt, G.; Runborg, O., A contribution to wavelet-based subgrid modeling, Appl. Comput. Harmon. Anal., 7, 2, 151-164 (1999) · Zbl 0937.65113
[3] Bai, Z.; Demmel, J.; Dongarra, J.; Ruhe, A.; van der Vorst, H., Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide (2000), SIAM: SIAM Philadelphia · Zbl 0965.65058
[4] Bensoussan, A.; Lions, J.-L.; Papanicolau, G., Asymptotic Analysis for Periodic Structures (1978), North-Holland: North-Holland Amsterdam
[5] Beylkin, G.; Coifman, R.; Rokhlin, V., Fast wavelet transforms and numerical algorithms I, Comm. Pure Appl. Math., 44, 141-183 (1991) · Zbl 0722.65022
[6] Beylkin, G.; Coult, N., A multiresolution strategy for reduction of elliptic PDE’s and eigenvalue problems, Appl. Comput. Harmon. Anal., 5, 129-155 (1998) · Zbl 0912.35013
[7] Brewster, M. E.; Beylkin, G., A multiresolution strategy for numerical homogenization, Appl. Comput. Harmon. Anal., 2, 327-349 (1995), PAM Report 187, 1994 · Zbl 0840.65047
[8] Y. Capdeboscq, M.S. Vogelius, Wavelet based homogenization of a 2 dimensional elliptic problem, Preprint, 2002; Y. Capdeboscq, M.S. Vogelius, Wavelet based homogenization of a 2 dimensional elliptic problem, Preprint, 2002
[9] A. Chertock, D. Levy, On wavelet-based numerical homogenization, SIAM J. Multiscale Math. Simul., in press; A. Chertock, D. Levy, On wavelet-based numerical homogenization, SIAM J. Multiscale Math. Simul., in press · Zbl 1079.65104
[10] N. Coult, A multiresolution strategy for homogenization of partial differential equations, Ph.D. thesis, University of Colorado, 1997; N. Coult, A multiresolution strategy for homogenization of partial differential equations, Ph.D. thesis, University of Colorado, 1997
[11] Wittum, G.; Wagner, C.; Kinzelbach, W., Schur-complement multigrid—A robust method for groundwater flow and transport problems, Numer. Math., 75, 523-545 (1997) · Zbl 0876.76066
[12] Daubechies, I., Ten Lectures on Wavelets, CBMS-NSF Series in Appl. Math. (1992), SIAM: SIAM Philadelphia · Zbl 0776.42018
[13] Dorobantu, M.; Engquist, B., Wavelet-based numerical homogenization, SIAM J. Numer. Anal., 35, 2, 540-559 (1998) · Zbl 0936.65135
[14] Gilbert, A. C., A comparison of multiresolution and classical one-dimensional homogenization schemes, Appl. Comput. Harmon. Anal., 5, 1, 1-35 (1998) · Zbl 0896.35016
[15] Gines, D. L.; Beylkin, G.; Dunn, J., LU factorization of non-standard forms and direct multiresolution solvers, Appl. Comput. Harmon. Anal., 5, 156-201 (1998), PAM Report 278, 1996 · Zbl 0914.65017
[16] Prasolov, V. V., Problems and Theorems in Linear Algebra, Translations of Mathematical Monographs, vol. 134 (1994), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0803.15001
[17] Santosa, F.; Symes, W., A dispersive effective medium for wave propagation in periodic composites, SIAM J. Appl. Math., 51, 984-1005 (1991) · Zbl 0741.73017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.