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Divergence-free wavelet bases on the hypercube: Free-slip boundary conditions, and applications for solving the instationary Stokes equations


Author: Rob Stevenson
Journal: Math. Comp. 80 (2011), 1499-1523
MSC (2010): Primary 35K99, 65T60, 65M12, 76D03
DOI: https://doi.org/10.1090/S0025-5718-2011-02471-3
Published electronically: February 28, 2011
MathSciNet review: 2785466
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Abstract: We construct wavelet Riesz bases for the usual Sobolev spaces of divergence free functions on $ (0,1)^n$ that have vanishing normals at the boundary. We give a simultaneous space-time variational formulation of the instationary Stokes equations that defines a boundedly invertible mapping between a Bochner space and the dual of another Bochner space. By equipping these Bochner spaces by tensor products of temporal and divergence-free spatial wavelets, the Stokes problem is rewritten as an equivalent well-posed bi-infinite matrix vector equation. This equation can be solved with an adaptive wavelet method in linear complexity with best possible rate, that, under some mild Besov smoothness conditions, is nearly independent of the space dimension. For proving one of the intermediate results, we construct an eigenfunction basis of the stationary Stokes operator.


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  • [CS10] N.G. Chegini and R.P. Stevenson.
    Adaptive wavelets schemes for parabolic problems: Sparse matrices and numerical results.
    Technical report, Korteweg-de Vries Institute, University of Amsterdam, 2010. To appear in SIAM J. Numer. Anal.
  • [Dah96] W. Dahmen.
    Stability of multiscale transformations.
    J. Fourier Anal. Appl., 2(4):341-362, 1996. MR 1395769 (97i:46133)
  • [Dau89] M. Dauge.
    Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. I. Linearized equations.
    SIAM J. Math. Anal., 20(1):74-97, 1989. MR 977489 (90b:35191)
  • [Dij09] T.J. Dijkema.
    Adaptive tensor product wavelet methods for solving PDEs.
    Ph.D. thesis, Utrecht University, 2009.
  • [DKU99] W. Dahmen, A. Kunoth, and K. Urban.
    Biorthogonal spline-wavelets on the interval: Stability and moment conditions.
    Appl. Comp. Harm. Anal., 6:132-196, 1999. MR 1676771 (99m:42046)
  • [DL92] R. Dautray and J.-L. Lions.
    Mathematical analysis and numerical methods for science and technology. Vol. 5,
    Springer-Verlag, Berlin, 1992.
    Evolution problems I. MR 1156075 (92k:00006)
  • [DP06] E. Deriaz and V. Perrier.
    Divergence-free and curl-free wavelets in two dimensions and three dimensions: Application to turbulent flows.
    J. Turbul., 7:Paper 3, 37 pp. (electronic), 2006. MR 2207365 (2006i:76080)
  • [DS09] M. Dauge and R.P. Stevenson.
    Sparse tensor product wavelet approximation of singular functions. SIAM J. Math. Anal., 42(5):2203-2228, 2010. MR 2729437
  • [KO76] R.B. Kellogg and J.E. Osborn.
    A regularity result for the Stokes equation in a convex polygon.
    J. Funct. Anal., 21:397-431, 1976. MR 0404849 (53:8649)
  • [LR92] P.G. Lemarié-Rieusset.
    Analyses multi-résolutions non-orthogonales, commutation entre projecteurs et dérivation et ondelettes vecteurs à divergence nulle.
    Rev. Mat. Iberoamericana, 8(2):221-237, 1992. MR 1191345 (94d:42044)
  • [LR02] D.-S. Lee and B. Rummler.
    The eigenfunctions of the Stokes operator in special domains. III.
    ZAMM Z. Angew. Math. Mech., 82(6):399-407, 2002. MR 1906228 (2003f:35232)
  • [MM09] M. Mitrea and S. Monniaux.
    On the analyticity of the semigroup generated by the Stokes operator with Neumann-type boundary conditions on Lipschitz subdomains of Riemannian manifolds.
    Trans. Amer. Math. Soc., 361(6):3125-3157, 2009. MR 2485421 (2010h:35078)
  • [Nit06] P.-A. Nitsche.
    Best $ N$-term approximation spaces for tensor product wavelet bases.
    Constr. Approx., 24(1):49-70, 2006. MR 2217525 (2007a:41032)
  • [SS09] Ch. Schwab and R.P. Stevenson.
    A space-time adaptive wavelet method for parabolic evolution problems.
    Math. Comp., 78:1293-1318, 2009. MR 2501051
  • [Ste08] R.P. Stevenson.
    Divergence-free wavelets on the hypercube.
    Technical report, Korteweg-de Vries Institute for Mathematics, 2008. Appl. Comput. Harmon. Anal., 30:1-19, 2011.
  • [Ste09] R.P. Stevenson.
    Adaptive wavelet methods for solving operator equations: An overview.
    In R.A. DeVore and A. Kunoth, editors, Multiscale, Nonlinear and Adaptive Approximation: Dedicated to Wolfgang Dahmen on the Occasion of his 60th Birthday, pages 543-598. Springer, 2009.
  • [Tem79] R. Temam.
    Navier-Stokes equations, volume 2 of Studies in Mathematics and its Applications.
    North-Holland Publishing Co., Amsterdam, revised edition, 1979.
    Theory and numerical analysis, With an appendix by F. Thomasset. MR 603444 (82b:35133)
  • [Wlo82] J. Wloka.
    Partielle Differentialgleichungen.
    B. G. Teubner, Stuttgart, 1982.
    Sobolevräume und Randwertaufgaben. MR 652934 (84a:35002)

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Additional Information

Rob Stevenson
Affiliation: Korteweg–de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands
Email: R.P.Stevenson@uva.nl

DOI: https://doi.org/10.1090/S0025-5718-2011-02471-3
Keywords: Divergence free wavelets, simultaneous space-time variational formulation, instationary Stokes equations, adaptive wavelet scheme, eigenfunction basis for stationary Stokes operator, elliptic regularity
Received by editor(s): March 9, 2010
Received by editor(s) in revised form: March 16, 2010
Published electronically: February 28, 2011
Article copyright: © Copyright 2011 American Mathematical Society

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