Interdisciplinary Applied Mathematics

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Now the approximation of u is given by uapprox = umn + h(umn), which is analogous to equation (17.25). In the actual process, first, one makes a standard Galerkin approximation using n nodes. Then the m-dimensional approximation of the inertial manifold, h, is calculated (see (Brown et al., 1990), for details) and used in equation (17.28) and in the expression for uapprox. This method can capture nonlinear behavior better than standard Galerkin methods, but it involves extra cost, since the inertial manifold needs to be computed at every integration step.

Postprocessed Galerkin Method

This method is computationally more efficient than the approximate inertial manifold theory (Garcia-Archilla et al., 1998). In this method, first the standard Galerkin method is used, and at time (t) only when some output is required, the variables are approximated by the inertial manifold. That is, the solution qm = qml is calculated from

q ml = P Lqml + P g(qmi),

which requires less effort than computing qm = qmn from

qmn    PLqmn + Pg(qmn + happrox(qmn ))

The final    approximate    solution    for    u,    uapprox,    is    computed    by    uapprox =

uml + happrox(uml), where happrox is the approximate inertial manifold. The computational cost is reduced greatly as a result of this simplification.

The concept of dynamic postprocessing has been introduced in (Margolin et al.,    2003),    for    highly    oscillatory    systems.    For    a variety    of    systems,    the

normal postprocessed Galerkin method has been found to be a very efficient technique for improving the accuracy of ordinary Galerkin/nonlinear Galerkin methods with very little extra computational cost. The normal postprocessed Galerkin methods are based on truncation analysis using asymptotic (in time) estimates for the low and high mode components, which hold    only when    the    solutions    are    on    or    near an    attractor.    As a    re

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