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I Vector-wavelet Galerkin projection of Navier-Stokes equation

  1. Aug 6, 2017 #1
    Hi,

    I am having a little trouble understanding a minor step in a paper by [V. Zimin and F. Hussain][1].

    They define a collection of divergence-free vector wavelets as

    $$\mathbf{v}_{N\nu n}(\mathbf{x}) = -\frac{9}{14}\rho^{1/2}_N \mathbf{e}_\nu\times\nabla_\mathbf{s}\bigg(\frac{\cos(s)-\cos(2s)}{s^2}\bigg),$$

    where

    $$ \rho_N = \frac{7\pi}{9}2^{3N} \qquad\text{and}\qquad \mathbf{s} = \pi2^N(\mathbf{x}-\mathbf{x}_{Nn}).$$

    Here does $N$ denote the "scale" of the wavelet.

    Zimin and Hussain then write the Galerkin projection of Navier-Stokes equation onto this divergence-free vector wavelet projection. The Laplacian term in NSE become

    $$ \sum_{\mu}\sum_{m}A_{N\mu m}\int \mathbf{v}_{N\nu n}\cdot\Delta\mathbf{v}_{N\mu m}\,d^3\mathbf{x}. $$

    But, when I try to write down this Galerkin projection I find that the Laplacian term becomes
    $$ \sum_{M}\sum_{\mu}\sum_{m}A_{M\mu m}\int \mathbf{v}_{N\nu n}\cdot\Delta\mathbf{v}_{M\mu m}\,d^3\mathbf{x}. $$

    This is almost, identical to Zimin and Hussain. But I have a additional sum over the scale-index $M$, and I can not seem to get rid of it.

    NB. My feeling is, that this has nothing to do with the explicit form of the divergence-free vector wavelets. But something to do with the fact that they are wavelets (and thus an orthonormal basis) and perhaps (but not so much) that they are divergence-free.

    [1]: http://dx.doi.org/10.1063/1.868669
     
  2. jcsd
  3. Aug 11, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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