Gibbs phenomena for $L^q$-best approximation in finite element spaces

Houston, Paul; Roggendorf, Sarah; van der Zee, Kristoffer G.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 1, pp. 177-211

Recent developments in the context of minimum residual finite element methods are paving the way for designing quasi-optimal discretization methods in non-standard function spaces, such as q-type Sobolev spaces. For q → 1, these methods have demonstrated huge potential in avoiding the notorious Gibbs phenomena, i.e., the occurrence of spurious non-physical oscillations near thin layers and jump discontinuities. In this work we provide theoretical results that explain some of these numerical observations. In particular, we investigate the Gibbs phenomena for q-best approximations of discontinuities in finite element spaces with 1 ≤ q < ∞. We prove sufficient conditions on meshes in one and two dimensions such that over- and undershoots vanish in the limit q → 1. Moreover, we include examples of meshes such that Gibbs phenomena remain present even for q = 1 and demonstrate that our results can be used to design meshes so as to eliminate the Gibbs phenomenon.

Classification : 65N30, 41A10
Keywords: Best approximation, Gibbs phenomenon, $$^q, finite elements

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