# Battling Gibbs Phenomenon: On Finite Element Approximations of Discontinuous Solutions of PDEs

## TL;DR

This paper investigates the Gibbs phenomenon in finite element approximations of discontinuous PDE solutions, analyzing how mesh alignment and element type affect overshoot behavior and accuracy.

## Contribution

It provides explicit analysis and calculations of overshoot values for different finite element approaches and mesh configurations, offering practical guidance for reducing Gibbs phenomenon effects.

## Key findings

- Piecewise discontinuous approximations perform well on aligned meshes.
- Non-matched meshes with discontinuous elements can still achieve accuracy with adaptive refinement.
- Overshoot values are explicitly calculated for typical cases, guiding mesh and element choice.

## Abstract

In this paper, we want to clarify the Gibbs phenomenon when continuous and discontinuous finite elements are used to approximate discontinuous or nearly discontinuous PDE solutions from the approximation point of view. For a simple step function, we explicitly compute its continuous and discontinuous piecewise constant or linear projections on discontinuity matched or non-matched meshes. For the simple discontinuity-aligned mesh case, piecewise discontinuous approximations are always good. For the general non-matched case, we explain that the piecewise discontinuous constant approximation combined with adaptive mesh refinements is a good choice to achieve accuracy without overshoots. For discontinuous piecewise linear approximations, non-trivial overshoots will be observed unless the mesh is matched with discontinuity. For continuous piecewise linear approximations, the computation is based on a "far-away assumption", and non-trivial overshoots will always be observed under regular meshes. We calculate the explicit overshoot values for several typical cases. Numerical tests are conducted for a singularly-perturbed reaction-diffusion equation and linear hyperbolic equations to verify our findings in the paper. Also, we discuss the $L^1$-minimization-based methods and do not recommend such methods due to their similar behavior to $L^2$-based methods and more complicated implementations.