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

Mathematics of Computation

Volumn 80, Issue 275, 2011, Pages 1499-1523

  Stevenson, Rob ^a  

^a UNIVERSITY OF AMSTERDAM   (Netherlands)

Author keywords

Adaptive wavelet scheme; Divergence free wavelets; Eigenfunction basis for stationary Stokes operator; Elliptic regularity; Instationary stokes equations; Simultaneous space time variational formulation

Indexed keywords

EID: 79955655611     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-2011-02471-3     Document Type: Article

References (18)

1. 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.
2. W. Dahmen. Stability of multiscale transformations. J. Fourier Anal. Appl., 2(4):341-362, 1996.
3. 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.
4. T.J. Dijkema. Adaptive tensor product wavelet methods for solving PDEs. Ph.D. thesis, Utrecht University, 2009.
5. 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.
6. R. Dautray and J.-L. Lions. Mathematical analysis and numerical methods for science and technology. Vol. 5, Springer-Verlag, Berlin, 1992. Evolution problems I.
7. 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.
8. M. Dauge and R.P. Stevenson. Sparse tensor product wavelet approximation of singular functions. SIAM J. Math. Anal., 42(5):2203-2228, 2010.
9. 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.
10. 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.
11. 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.
12. 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.
13. P.-A. Nitsche. Best N-term approximation spaces for tensor product wavelet bases. Constr. Approx., 24(1):49-70, 2006.
14. Ch. Schwab and R.P. Stevenson. A space-time adaptive wavelet method for parabolic evolution problems. Math. Comp., 78:1293-1318, 2009.
15. 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.
16. 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.
17. 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.
18. J. Wloka. Partielle Differentialgleichungen. B. G. Teubner, Stuttgart, 1982. Sobolevräume und Randwertaufgaben.