Constraints for eliminating the Gibbs phenomenon in finite element approximation spaces

TL;DR

This paper introduces a set of functionals and constraints for finite element approximation spaces to effectively eliminate or reduce the Gibbs phenomenon, improving the accuracy of numerical solutions near discontinuities.

Contribution

It proposes a novel approach using functionals and constraints to eliminate the Gibbs phenomenon in finite element methods, applicable to both continuous and discontinuous approximations.

Findings

Complete elimination of over- and undershoot in 1D continuous approximations

Significant suppression of oscillations in higher-dimensional discontinuous approximations

New functional-based constraints effectively mitigate spurious oscillations

Abstract

One of the major challenges in finite element methods is the mitigation of spurious oscillations near sharp layers and discontinuities known as the Gibbs phenomenon. In this article, we propose a set of functionals to identify spurious oscillations in best approximation problems in finite element spaces. Subsequently, we adopt these functionals in the formulation of constraints in an effort to eliminate the Gibbs phenomenon. By enforcing these constraints in best approximation problems, we can entirely eliminate over- and undershoot in one dimensional continuous approximations, and significantly suppress them in one- and higher-dimensional discontinuous approximations.