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 Lq-best approximation of
discontinuities in finite element spaces with 1β€q<β. 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 L1 as a solution space in connection with suitably designed
meshes