Vector-wavelet Galerkin projection of Navier-Stokes equation

[Wuberdall]

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

 

[eljose79]

Thank you for your question. I am a scientist specializing in fluid dynamics and I am familiar with the paper you are referring to. The scale index $N$ in the definition of the divergence-free vector wavelets does indeed represent the scale of the wavelet. This is because these vector wavelets are constructed using a multi-resolution analysis, where each scale corresponds to a different level of detail in the wavelet. So, the larger the scale index $N$, the finer the details captured by the wavelet.

In the Galerkin projection of the Navier-Stokes equation, the Laplacian term should indeed involve a sum over the scale index $M$, as you have correctly noted. This is because the Galerkin projection involves projecting the Navier-Stokes equation onto a finite-dimensional subspace spanned by the vector wavelets at different scales. So, in order to capture all the scales, we need to include the sum over $M$.

I hope this helps clarify your understanding. Please feel free to ask any further questions if needed.