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

TL;DR

This paper investigates Gibbs phenomena in $L^q$-best approximation within finite element spaces, demonstrating potential elimination of oscillations near discontinuities as $q$ approaches 1, highlighting $L^1$'s promise for improved solutions.

Contribution

It explores the use of $L^q$ spaces for finite element approximation, showing how $L^1$ can reduce Gibbs phenomena with appropriate mesh design.

Findings

Gibbs phenomenon can be eliminated as $q$ approaches 1 on certain meshes.

$L^1$ space shows potential for better handling of discontinuities.

Mesh design plays a crucial role in controlling oscillations.

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.