On the robustness of the dampedV-cycle of the wavelet frequency decomposition multigrid method

The dampedV-cycle of the wavelet variation of the “Frequency decomposition multigrid method” of Hackbusch [Numer. Math.56, pp. 229–245 (1989)] is considered. It is shown that the convergence speed under sufficient damping is not affected by the presence of anisotropy but still depends on the number of levels. Our analysis is based on properties of wavelet packets which are supplied and proved. Numerical approximations to the speed of convergence illustrate the theoretical results.ZusammenfassungWir betrachten den gedämpftenV-Zyklus für die Wavelet-Variante der “Frequenzzerlegungs-Multigridmethode” von Hackbusch [Numer. Math.56, 229–245 (1989)]. Es wird gezeigt, daß die Konvergenzgeschwindigkeit bei hinreichender Dämpfung durch Anisotropie nicht beeinflußt wird, aber noch von der Anzahl des Niveaus abhängt. Unsere Analyse beruht auf Eigenschaften von Wavelet-Paketen, die formuliert und bewiesen werden. Numerische Schätzungen der Konvergenzgeschwindigkeit erläutern die theoretischen Ergebnisse.

[1]  I. Daubechies,et al.  Frames in the Bargmann Space of Entire Functions , 1988 .

[2]  Yves Meyer.  Wavelets - algorithms & applications , 1993 .

[3]  M. Victor Wickerhauser,et al.  Wavelets: Algorithms and Applications (Yves Meyer) , 1994, SIAM Rev..

[4]  Junping Wang,et al.  Convergence analysis without regularity assumptions for multigrid algorithms based on SOR smoothing , 1992 .

[5]  J. Pasciak,et al.  Convergence estimates for product iterative methods with applications to domain decomposition , 1991 .

[6]  G. Beylkin.  On the representation of operators in bases of compactly supported wavelets , 1992 .

[7]  Yves Meyer,et al.  Size properties of wavelet packets , 1992 .

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

[9]  R. Bank,et al.  The hierarchical basis multigrid method , 1988 .

[10]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.

[11]  En-Bing Lin,et al.  Coiflet interpolation and approximate solutions of elliptic partial differential equations , 1997 .

[12]  S. Mallat.  Multiresolution approximations and wavelet orthonormal bases of L^2(R) , 1989 .

[13]  I. Daubechies.  Orthonormal bases of compactly supported wavelets , 1988 .

[14]  Andreas Rieder,et al.  A Wavelet Approach to Robust Multilevel Solvers for Anisotropic Elliptic Problems , 1994 .

[15]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[16]  T. Eirola.  Sobolev characterization of solutions of dilation equations , 1992 .

[17]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[18]  Jinchao Xu,et al.  Convergence estimates for multigrid algorithms without regularity assumptions , 1991 .

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