A dichotomy of finite element spaces and its application to an energy-conservative scheme for the regularized long wave equation

TL;DR

This paper investigates the intrinsic difference in convergence behavior of finite element spaces based on polynomial degree, revealing a super-approximation property for odd degrees and applying this insight to develop an energy-conservative scheme for the regularized long wave equation.

Contribution

The work uncovers the fundamental reason behind the odd-even convergence discrepancy in finite element spaces and applies this understanding to design an energy-conservative numerical scheme for a nonlinear wave equation.

Findings

Optimal convergence with odd polynomial degrees due to super-approximation property.

Reduced accuracy with even polynomial degrees linked to finite element structure.

The proposed scheme conserves mass and energy, with high-accuracy impulse approximation and proven error bounds.

Abstract

Certain energy-conservative Galerkin discretizations for nonlinear dispersive wave equations have revealed an unusual convergence behavior: optimal convergence is attained when continuous Lagrange finite element spaces of odd polynomial degree are employed, whereas the use of even-degree polynomials leads to reduced accuracy. The present work demonstrates that this behavior is intrinsic to the structure of the finite element spaces themselves. In particular, it is shown to be closely connected to the standard $L^2$-projection of derivatives, which possesses a super-approximation property exclusively for odd polynomial degrees. We also examine the implications of this feature for an energy-conservative Galerkin approximation of the regularized long-wave equation where the energy is a cubic functional. Although the resulting scheme conserves both mass and energy, we further show that the impulse is approximated with high accuracy, and we establish {\em a priori} error bounds for the associated semi-discrete formulation.