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Wavelet preconditioning of the Stokes problem in (psgr, ohgr) formulation

Pascal JolyContact Information and Roland MassonContact Information

(1)  Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie, tour 55-65, 5ème étage, 4 place Jussieu, F-75252 Paris Cedex 05, France
(2)  Département Informatique Scientifique et Mathématiques Appliquées, Institut Français du Pétrole, F-92500 Rueil Malmaison Cedex, France

Abstract  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 (psgr, ohgr) formulation using a diagonal preconditioning in wavelet basis for the H1/2(part OHgr) boundary operator that relates the trace of partn psgr to the trace of ohgr.

psi-omega - wavelets - multilevel preconditioners - negative smoothness - elliptic operators - 42C15 - 65N55

This revised version was published online in June 2006 with corrections to the Cover Date.

Contact Information Pascal Joly
Email: joly@ann.jussieu.fr

Contact Information Roland Masson
Email: roland.masson@ifp.fr

References

[1]. L. Andersson, N. Hall, B. Jawerth and G. Peters, Wavelets on closed subsets of the real line, in: Recents Advances in Wavelets Analysis, eds. L.L. Schumaker and G. Webb (Academic Press, New York, 1993) pp. 1-61.
 
[2]. M. Bercovier and A. Engelman, A finite element method for the numerical solution of viscous incompressible flows, J. Comput. Phys. 30 (1979) 181-201.
MATH AMS CrossRef
 
[3]. C. Canuto, A. Tabacco and K. Urban, Wavelet element method. Part 1: Construction and analysis, Dip. di Matematica, Politecnico di Torino, Rapporto Interno No 13 (1997), submitted to Appl. Comput. Harmon. Anal.
 
[4]. A. Cohen, Wavelet methods in numerical analysis, in: Handbook of Numerical Analysis, Vol. VII, eds. P.G. Ciarlet and J.L. Lions (Elsevier Science/North-Holland, Amsterdam).
 
[5]. A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet methods for elliptic operator equations-convergence rates, preprint (1998).
 
[6]. A. Cohen, I. Daubechies and J. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. 45 (1992) 485-560.
MATH AMS
 
[7]. A. Cohen, I. Daubechies and P. Vial, Wavelets on the interval and fast wavelet transforms, Appl. Comput. Harmon. Anal. 1 (1993) 54-81.
MATH AMS CrossRef
 
[8]. A. Cohen, L.M. Echeverry and Q. Sun, Finite element based biorthogonal wavelets on plane polygones, Preprint LAN, Université Pierre et Marie Curie (1999).
 
[9]. A. Cohen and R. Masson, Wavelet adaptive methods for 2nd order elliptic problems, Preprint No. 97036, Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie (1997), to appear in SIAM J. Sci. Comput.
 
[10]. A. Cohen and R. Masson, Wavelet adaptive methods for 2nd order elliptic problems, domain decomposition and boundary conditions, Preprint No. 98007, Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie (1998), accepted in Numerische Mathematik.
 
[11]. W. Dahmen, Wavelent and multiscale methods for operator equations, in: Acta Numerica (Cambridge Univ. Press, Cambridge, 1997) pp. 55-228.
 
[12]. W. Dahmen and A. Kunoth, Multilevel preconditioning, Numer. Math. 63 (1992) 315-344.
MATH AMS SpringerLink
 
[13]. W. Dahmen, A. Kunoth and K. Urban, Spline biorthogonal wavelet on the interval, stability and moment conditions, Preprint No 129, IGPM, RWTH Aachen (1996), to appear in Appl. Comput. Harmon. Anal. (1999).
 
[14]. W. Dahmen and R. Schneider, Composite wavelet bases for operator equations, Publication No 133, Institut für Geometrie und Praktische Mathematik, RWTH Aachen (1996).
 
[15]. W. Dahmen and R. Schneider, Wavelets on manifolds I: Construction and domain decomposition, Preprint, Institut für Geometrie und Praktische Mathematik, RWTH Aachen (1997), to appear in SIAM J. Math. Anal.
 
[16]. V. Giraud and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Applications, Springer Series in Computational Mathematics (Springer, Berlin, 1986).
 
[17]. R. Glowinski and O. Pironneau, Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem, SIAM Rev. 21(2) (1979) 167-212.
MATH AMS CrossRef
 
[18]. Grivet and A. Tabacco, Biorthogonal wavelets on the interval with optimal support properties, Preprint, Dipartimento di Matematica, Politecnico di Torino (1998).
 
[19]. S. Jaffard, Wavelets methods for fast resolution of elliptic problems, SIAM J. Numer. Anal. 29 (1992) 965-986.
MATH AMS CrossRef
 
[20]. A. Jouini and P.G. Lemarie-Rieusset, Ondelettes sur un ouvert borné du plan, Preprint, Université d'Orsay (1992).
 
[21]. Y. Maday, V. Perrier and J.C. Ravel, Adaptativité dynamique sur bases d'ondelettes pour l'approximaton d'équations aux dérivées partielles, C. R. Acad. Sci. Paris Série I Math. I (1991) 405-410.
MATH AMS
 
[22]. R. Masson, Biorthogonal spline wavelets on the interval for the resolution of boundary problems, Math. Models Methods Appl. Sci. 6(6) (1996) 749-791.
MATH AMS CrossRef
 
[23]. R. Stevenson, Piecewise linear pre-wavelets on non-uniform meshes, Report 9701, University of Nijmegen, submitted to Proceedings EMG'96.
 
[24]. R.A. Sweet, A cyclic reduction algorithm for solving block tridiagonal systems of arbitrary dimension, SIAM J. Numer. Anal. 14 (1977) 706-720.
MATH AMS CrossRef
 
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