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

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- by Rob Stevenson PDF
- Math. Comp.
**80**(2011), 1499-1523 Request permission

## 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.## References

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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
- Email: R.P.Stevenson@uva.nl
- Received by editor(s): March 9, 2010
- Received by editor(s) in revised form: March 16, 2010
- Published electronically: February 28, 2011
- © Copyright 2011 American Mathematical Society
- 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
- MathSciNet review: 2785466