# enabling a wavelet-Galerkin turbulent flow field representation with expansion coefficients in identical physical & wavelet space

“*When Fourier meets Navier*” in the physical H(-1/2) Hilbert space there is a global unique solution of the corresponding weak variation equation of the non-linear, non-stationary Navier-Stokes equations ("solution"). The corresponding numerical approximation method is the Ritz-Galerkin method which is usually equipped with finite element approximation spaces. In case of negative scaled Hilbert spaces this is called boundary element method related to the underlying singular equation representation. The finite (boundary) element approximation properties face some challenges in case of non-linearity and/or non-periodic boundary conditions. The wavelet extension method is used to represent functions that have discontinuities and sharp peaks, and for accurately deconstructing and reconstructing finite, non-periodic and/or non-stationary signals.

The wavelet theory is established in the Fourier Hilbert space framework. In order to apply the Calderón inverse formula in a Hilbert scale Framework it requires the so-called admissibility condition defining a wavelet analyzing a (signal) function. "*A wavelet synthesis can be performed locally as opposed to the Fourier transform which is inherently nonlocal due to the space-filling nature of the trigonometric functions*" (FaM1). So the question is, how to bundle both necessary concepts into one applying it to corresponding "right" physical NSE variation representation with related energy Hilbert space.

Here we go with a collection of useful tool sets: