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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?

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Take a look at some of the papers by Jean-Luc Guermond over the past 10 or so years that deal with solving PDEs my minimizing the residual in the L1 norm. His website is here.

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