Wavelet preconditioning of the Stokes problem in ψ–ω formulation

The diagonal preconditioning in wavelet basis enables one to obtain an optimal preconditioner for Galerkin discretizations of elliptic operators in Sobolev norms of both positive and negative smoothness. We develop these techniques in order to solve efficiently the bi-Laplacian or the bidimensional Stokes problem in ψ–ω formulation using a diagonal preconditioning in wavelet basis for the H−1/2(∂Ω) boundary operator that relates the trace of ∂nψ to the trace of ω.

[1]  M. Bercovier,et al.  A finite element for the numerical solution of viscous incompressible flows , 1979 .

[2]  Claudio Canuto,et al.  The wavelet element method. Part I: Construction and analysis. , 1997 .

[3]  W. Dahmen,et al.  Biorthogonal Spline Wavelets on the Interval—Stability and Moment Conditions , 1999 .

[4]  W. Dahmen,et al.  Multilevel preconditioning , 1992 .

[5]  Albert Cohen,et al.  Wavelet methods in numerical analysis , 2000 .

[6]  R. Sweet.  A Cyclic Reduction Algorithm for Solving Block Tridiagonal Systems of Arbitrary Dimension , 1977 .

[7]  Wolfgang Dahmen,et al.  Composite wavelet bases for operator equations , 1999, Math. Comput..

[8]  Y. Maday,et al.  ADAPTATIVITE DYNAMIQUE SUR BASES D'ONDELETTES POUR L'APPROXIMATION D'EQUATIONS AUX DERIVEES PARTIELLES , 1991 .

[9]  Wolfgang Dahmen,et al.  Wavelets on Manifolds I: Construction and Domain Decomposition , 1999, SIAM J. Math. Anal..

[10]  Wolfgang Dahmen,et al.  Adaptive wavelet methods for elliptic operator equations: Convergence rates , 2001, Math. Comput..

[11]  I. Daubechies,et al.  Wavelets on the Interval and Fast Wavelet Transforms , 1993 .

[12]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[13]  W. Dahmen.  Wavelet and multiscale methods for operator equations , 1997, Acta Numerica.

[14]  R. Glowinski,et al.  Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem , 1977 .

[15]  Roland Masson,et al.  BIORTHOGONAL SPLINE WAVELETS ON THE INTERVAL FOR THE RESOLUTION OF BOUNDARY PROBLEMS , 1996 .

[16]  Rob Stevenson.  Piecewise linear (pre-)wavelets on non-uniform meshes , 1998 .

[17]  F. Thomasset.  Finite element methods for Navier-Stokes equations , 1980 .

[18]  I. Daubechies,et al.  Biorthogonal bases of compactly supported wavelets , 1992 .

[19]  S. Jaffard.  Wavelet methods for fast resolution of elliptic problems , 1992 .