| Summary: | 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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