Gibbs phenomena for Lq-best approximation in finite element spaces

Houston, Paul; Roggendorf, Sarah; Van Der Zee, Kristoffer G

Abstract

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 L 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 L 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.

Citation

Houston, P., Roggendorf, S., & Van Der Zee, K. G. (2022). Gibbs phenomena for Lq-best approximation in finite element spaces. ESAIM: Mathematical Modelling and Numerical Analysis, 56(1), 177-211. https://doi.org/10.1051/m2an/2021086

• Journal Article Type: Article
• Acceptance Date: Dec 16, 2021
• Online Publication Date: Feb 7, 2022
• Publication Date: Feb 7, 2022
• Deposit Date: Dec 17, 2021
• Publicly Available Date: Feb 7, 2022
• Journal: ESAIM: Mathematical Modelling and Numerical Analysis
• Print ISSN: 0764-583X
• Electronic ISSN: 2804-7214
• Peer Reviewed: Peer Reviewed
• Volume: 56
• Issue: 1
• Pages: 177-211
• DOI: https://doi.org/10.1051/m2an/2021086
• Keywords: Applied Mathematics; Modeling and Simulation; Numerical Analysis; Analysis; Computational Mathematics
• Public URL: https://nottingham-repository.worktribe.com/output/7023257
• Publisher URL: https://www.esaim-m2an.org/component/article?access=doi&doi=10.1051/m2an/2021086
• Additional Information: The original publication is available at https://www.esaim-m2an.org/component/article?access=doi&doi=10.1051/m2an/2021086