# L1 functional setting for Navier-Stokes with finite elements

Typical finite element problems assume $L^2$ which is a Hilbert space, but I've heard that $L^1$ for Navier-Stokes results in less overshoots/undershoots, but $L^1$ is not Hilbert. The dual space for $L^1$ is $L^\infty$, so it seems that we might want some sort of Petrov-Galerkin formulation. But practically, do these different spaces result in different choices for the basis functions?