On certain 1D and 2D meshes, the Lq-best approximation of a discontinuity in piecewise-linear finite element spaces has over/undershoots that vanish as q tends to 1, while on other meshes they persist even at q=1.
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Gibbs Phenomena for $L^q$-Best Approximation in Finite Element Spaces -- Some Examples
On certain 1D and 2D meshes, the Lq-best approximation of a discontinuity in piecewise-linear finite element spaces has over/undershoots that vanish as q tends to 1, while on other meshes they persist even at q=1.