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Gibbs Phenomena for $L^q$-Best Approximation in Finite Element Spaces -- Some Examples

Authors :
Houston, Paul
Roggendorf, Sarah
van der Zee, Kristoffer G.
Source :
ESAIM: Mathematical Modelling and Numerical Analysis 2022
Publication Year :
2019

Abstract

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 $L^q$-best approximation of discontinuities in finite element spaces with $1\leq q<\infty$. 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 $L^1$ as a solution space in connection with suitably designed meshes.

Details

Database :
arXiv
Journal :
ESAIM: Mathematical Modelling and Numerical Analysis 2022
Publication Type :
Report
Accession number :
edsarx.1909.00658
Document Type :
Working Paper
Full Text :
https://doi.org/10.1051/m2an/2021086