Volume 56, Number 1, January-February 2022
|Page(s)||177 - 211|
|Published online||07 February 2022|
Gibbs phenomena for Lq-best approximation in finite element spaces*
School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, UK
2 Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK
* Corresponding author: firstname.lastname@example.org
Accepted: 16 December 2021
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.
Mathematics Subject Classification: 65N30 / 41A10
Key words: Best approximation / Gibbs phenomenon / Lq / finite elements
© The authors. Published by EDP Sciences, SMAI 2022
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