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

Technical Summary: A Dichotomy of Finite Element Spaces and Its Application to an Energy-Conservative Scheme for the Regularized Long Wave Equation

Problem Statement
The paper addresses a specific phenomenon observed in energy-conservative Galerkin discretizations of nonlinear dispersive wave equations: an unusual dichotomy in convergence behavior based on the polynomial degree of the finite element space. Specifically, optimal convergence rates are achieved when using continuous Lagrange finite element spaces of odd polynomial degrees, whereas even-degree polynomials yield reduced convergence rates. The authors aim to demonstrate that this behavior is intrinsic to the structure of the finite element spaces themselves, rather than an artifact of specific numerical schemes. Furthermore, the paper investigates the implications of this property for the Regularized Long Wave (RLW) equation (also known as the Benjamin–Bona–Mahony or BBM equation), a fundamental model for long surface water waves that possesses mass, impulse, and energy conservation laws.

Methodology
The analysis is conducted within the framework of one-dimensional continuous Lagrange finite element spaces ($V^k_h$) defined on a uniform periodic grid. The core of the methodology involves a direct-sum decomposition of the finite element space $V^k_h = S^1_h \oplus S^2_h$, where $S^1_h$ consists of vertex-based functions and $S^2_h$ comprises bubble functions.

1. Analysis of $L^2$-Projections: The authors rigorously analyze the $L^2$-projection $P$ of derivatives. They prove that for sufficiently smooth periodic functions, the error term $\|P[(Pu - u)_x]\|$ exhibits a "super-approximation" property exclusively for odd polynomial degrees ($k = 2\kappa + 1$), achieving an order of $O(h^{k+1})$. In contrast, for even degrees ($k = 2\kappa$), the error remains at the standard optimal order $O(h^k)$. This distinction is derived through detailed estimates of the projections onto the vertex and bubble subspaces, utilizing properties of orthogonal polynomials and specific basis functions (such as $\psi$ and $\phi_j$).
2. Semi-Discrete Formulation: A modified Galerkin finite element method is proposed for the RLW equation. This scheme is designed to conserve both mass and a cubic energy functional. The semi-discrete formulation is expressed as a system of ordinary differential equations involving the $L^2$-projection of the nonlinear and dispersive terms.
3. Error Analysis: Theoretical a priori error bounds are established for the semi-discrete formulation. The proof relies on the dichotomy property of the $L^2$-projection of derivatives established in Section 2. The analysis distinguishes between odd and even $k$ to derive the specific convergence rates for the solution $u$ and its derivative $u_x$.
4. Impulse Approximation: While the proposed scheme conserves mass and energy, it does not strictly conserve the impulse functional. The authors analyze the accuracy of the impulse approximation, showing it is preserved with high accuracy dependent on the polynomial degree.
5. Fully Discrete Scheme: To integrate the semi-discrete system in time while preserving the energy functional, the authors employ Relaxation Runge–Kutta methods. These methods adjust the time step via a relaxation parameter $\gamma_n$ to ensure exact conservation of the discrete energy.

Key Contributions and Results

• Theoretical Dichotomy: The primary contribution is the proof that the super-approximation property of the $L^2$-projection of derivatives is exclusive to odd-degree Lagrange finite element spaces. For odd $k$, $\|P[(Pu - u)_x]\| = O(h^{k+1})$; for even $k$, the bound is $O(h^k)$.
• Convergence Rates for RLW: Applying this property to the energy-conservative scheme for the RLW equation, the authors establish that:
  • For odd $k$: The numerical solution converges with optimal order $O(h^{k+1})$, and the derivative approximation $u_x$ also converges with order $O(h^{k+1})$ (super-convergence).
  • For even $k$: The convergence is suboptimal, with the solution converging at $O(h^k)$ and the derivative at $O(h^{k-1})$.
• Impulse Conservation: The paper demonstrates that while the modified scheme does not strictly conserve the impulse functional, the error in the impulse approximation is of order $O(h^{2k})$ for odd $k$ and $O(h^{2k-2})$ for even $k$.
• Numerical Validation: Extensive numerical experiments confirm the theoretical predictions. Convergence rates for the error $\|P[(Pu - u)_x]\|$ and the RLW solution match the derived bounds. Additionally, simulations show that mass and energy are conserved up to machine precision, while impulse errors remain small and follow the predicted convergence patterns.

Significance
The paper claims that the observed dichotomy in convergence behavior for energy-conservative schemes is not a numerical anomaly but a fundamental property of the finite element spaces used. By linking the convergence rate directly to the parity of the polynomial degree via the $L^2$-projection of derivatives, the work provides a theoretical explanation for previous empirical observations in the literature (e.g., [28]). The results suggest that for problems involving cubic energy functionals and derivative projections, the choice of odd-degree polynomials is not merely a heuristic but a necessity for achieving optimal accuracy. The proposed energy-conservative scheme for the RLW equation serves as a concrete application, demonstrating that high-order accuracy and conservation of fundamental physical quantities (mass and energy) can be simultaneously achieved, provided the appropriate finite element space is selected.