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On approximate inertial manifolds for the Navier-Stokes equations using finite elements Walsh, Owen


The nonlinearity in the Navier-Stokes equations couples the large and small scales of motion in turbulent flow. The nonlinear Galerkin method (NGM) consists of inserting into the equation for the large scale motion the small scale motion as determined by an “approximate inertial manifold”. Despite the conceptual appeal of this idea, its theoretic- cal justification has been recently thrown into question. However, its actual performance as a computational method has remained largely untested. Temam and collaborators have reported a 50% speed p in their spectral code for spatially periodic flow but their experiments have been recently criticized. In any case, spatially periodic computations are of little practical use. The aim of this thesis has been to test the NGM in the more practical context of the finite element method. Using finite elements, there is ambiguity and difficulty because the coarse grid has no natural supplementary space. We analyze a family of supplementary spaces and it is found that the quality of the asymptotic error estimates depends on the choice. Choosing the space by the L²-projection, we prove that the resulting approximation is “asymptotically good”. These results extend and improve upon recent error estimates of Marion and collaborators. For any other choice, the estimates are weaker and if -- as we suspect — they are optimal it seems possible that the NGM may actually decrease the accuracy of calculations. We also analyzed a variant of the NGM that we call “microscale linearization” (MSL). We prove that the MSL is “asymptotically good” for any member of this family of supplementary spaces. Turning to calculations, choosing the supplementary space by the Ritz projection, we implemented the NGM by modifying a 2-D Navier Stokes code of Turek; it performed very poorly. We implemented a variant of the MSL. It performed better, but still not as well as the original code. We sought a further understanding of these results by considering the 1-D Burgers equation. In conclusion, we find no numerical evidence that these methods are better than the standard finite element method. In fact, unless the coarse mesh is itself very fine, all versions performed poorly.

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