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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
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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Additional Information

Rob Stevenson
Affiliation: Korteweg–de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands
MR Author ID: 310898

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