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00876674 Urban, Karsten Multiscale methods for the Stokes problem and adapted wavelet bases. (Multiskalenverfahren f\"ur das Stokes-Problem und angepa{\ss}te Wavelet-Basen.) Aachener Beitr\"age zur Mathematik. 15. Aachen: Verlag der Augustinus Buchh. v, 223 p. (RWTH Aachen) (1995). 1995 Aachen: Verlag der Augustinus Buchh. DE stability Ladyshenskaja-Babu\v{s}ka-Brezzi condition driven-cavity problem multiscale methods Stokes equations adapted local wavelet bases Riesz basis incompressible Navier-Stokes equations divergence-free formulation local divergence-free wavelet bases mixed problems Summary: This thesis is concerned with the development and examination of multiscale methods for the Stokes equations and with the construction of adapted local wavelet bases for the Stokes problem. In general, multiscale methods are numerical schemes, where a sequence of nested scales is used for the approximation of the solution. The initial solution can be computed efficiently on a coarse scale. To obtain a more exact approximation one uses the relationship between different scales in such a way that finer details are added to the solution. This technique may lead to efficient numerical schemes. The claim of efficiency of those kinds of schemes implies some necessary conditions on the corresponding function system, that is, the basis functions have to form a Riesz basis for the particular function space. The Stokes equations are the linear, stationary part of the incompressible Navier-Stokes equations. They describe the flow of a viscous fluid in a domain $\Omega\subseteq \bbfR^n$. There exist at least two weak formulations for the Stokes problem which are used for the numerical modelling of this system of vector-valued partial differential equations. In the divergence-free formulation the incompressibility constraint, which is equivalent to the condition that the velocity field is divergence-free, is incorporated to the test- and trial-spaces. Hence, one needs divergence-free functions. Therefore, we construct local divergence-free wavelet bases. In particular, we show that the constructed wavelets form a Riesz basis for the space of divergence-free vector fields. The mixed formulation of the Stokes problem leads to a saddlepoint problem. For the stable numerical solvability the trial spaces must satisfy the Ladyshenskaja-Babu\v{s}ka-Brezzi-(LBB)-condition. We construct multiscale spaces that fulfill the LBB-condition. As a consequence of this construction, we obtain a stable wavelet decomposition of the space ${\bold H}(\text{div}; \Omega)^n$, which plays an important role for mixed problems. Several examples of the constructed wavelets are given. The numerical schemes are tested on the Driven-Cavity problem. This problem describes the flow of a medium over a box. We present computations for both weak formulations in two as well as three spatial dimensions. The associated software is documented in detail. It is natural to ask for extensions of the above results to Navier-Stokes equations. We consider the time dependent Stokes problem and show some possible strategies for the nonlinear case.