  Recent developments in the context of minimum residual finite element methods
are paving the way for designing finite element methods in non-standard
function spaces. This, in particular, permits the selection of a solution space
in which the best approximation of the solution has desirable properties. One
of the biggest challenges in designing finite element methods are non-physical
oscillations near thin layers and jump discontinuities. In this article we
investigate Gibbs phenomena in the context of $L^q$-best approximation of
discontinuities in finite element spaces with $1\leq q<\infty$. Using carefully
selected examples, we show that on certain meshes the Gibbs phenomenon can be
eliminated in the limit as $q$ tends to $1$. The aim here is to show the
potential of $L^1$ as a solution space in connection with suitably designed
meshes.
