Gibbs Phenomena for $L^q$-Best Approximation in Finite Element Spaces -- Some Examples
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 -best approximation of discontinuities in finite element spaces with . Using carefully selected examples, we show that on certain meshes the Gibbs phenomenon can be eliminated in the limit as tends to . The aim here is to show the potential of as a solution space in connection with suitably designed meshes.
Cite
@article{arxiv.1909.00658,
title = {Gibbs Phenomena for $L^q$-Best Approximation in Finite Element Spaces -- Some Examples},
author = {Paul Houston and Sarah Roggendorf and Kristoffer G. van der Zee},
journal= {arXiv preprint arXiv:1909.00658},
year = {2022}
}