Interdisciplinary Applied Mathematics

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1.    Approximate inertial manifold theory.


2.    Postprocessed Galerkin method.


Application of Karhunen-Loeve decomposition in nonlinear Galerkin methods has been presented in (Bangia et al., 1997). The dynamics of incompressible Navier-Stokes flow in a spatially periodic array of cylinders in a channel (for a mixing application) have been investigated using this method.


Approximate Inertial Manifold Theory


Equation (17.24) can be rewritten in the form (Steindl and Troger, 2001)


it c = P Luc + P g(uc + us),


(17.26)


us = QLus + Qg(uc + us),


(17.27)

by decomposing E = Ec ® Es, where Ec is finite-dimensional and Es is closed. This decomposition is achieved by defining the projection P onto Ec along    Es,    giving    uc    = Pu G    Ec    and    us    =    Qu    G    Es,    where Q =


I — P (see (Troger and Steindl, 1991), for details). In the standard Galerkin approximation of equation (17.24), from the eigenfunctions of L, m modes are selected, equation (17.27) is completely ignored, and us is set to be zero in equation (17.26) to obtain


u ml = P Lumi + Pg(umi),


where the index l denotes linear approximation. The influence of fast dynamics on the slow (essential) dynamics is completely ignored. Sometimes, a much better approximation is obtained if it is assumed that equation (17.24) has an inertial manifold of dimension m that can be realized as the graph of    a function    h : PE ^ QE,    or in    other    words,    us    =    h(umn).    The


projection of the inertial form onto PE is then given by


umn = P Lumn + P g(umn + h(umn))*    (17.28)

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