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Damped energy-norm a posteriori error estimates for fully discrete approximations of the wave equation using C2-reconstructions with the leapfrog scheme

T. Chaumont-Frelet, A. Ern

TL;DR

This work develops a rigorous a posteriori error framework for the fully discrete wave equation, discretized by continuous FEM in space and the explicit leapfrog scheme in time. The authors introduce two time reconstructions, a $C^0$ piecewise-quadratic $u_{h au}$ and a $C^2$ piecewise-quartic $w_{h au}$, to rewrite the discrete scheme and derive a damped-energy error estimator that is proven reliable and efficient. The analysis combines low-/high-frequency decomposition via Laplace transform with a data-oscillation term and a computable time-reconstruction error, yielding asymptotically refined upper and lower bounds in the damped energy norm. Numerical experiments in one dimension illustrate robustness, effectiveness of the estimator, and the accuracy of the reconstructions for standing and propagating waves. The results provide a practical tool for certifying fully discrete leapfrog solutions in wave-dominated simulations, under a fixed time step and mesh, using a damped energy framework.

Abstract

We derive a posteriori error estimates for the the scalar wave equation discretized in space by continuous finite elements and in time by the explicit leapfrog scheme. Our analysis combines the idea of invoking extra time-regularity for the right-hand side, as previously introduced in the space semi-discrete setting, with a novel, piecewise quartic, globally twice-differentiable time-reconstruction of the fully discrete solution. Our main results show that the proposed estimator is reliable and efficient in a damped energy norm. These properties are illustrated in a series of numerical examples.

Damped energy-norm a posteriori error estimates for fully discrete approximations of the wave equation using C2-reconstructions with the leapfrog scheme

TL;DR

This work develops a rigorous a posteriori error framework for the fully discrete wave equation, discretized by continuous FEM in space and the explicit leapfrog scheme in time. The authors introduce two time reconstructions, a piecewise-quadratic and a piecewise-quartic , to rewrite the discrete scheme and derive a damped-energy error estimator that is proven reliable and efficient. The analysis combines low-/high-frequency decomposition via Laplace transform with a data-oscillation term and a computable time-reconstruction error, yielding asymptotically refined upper and lower bounds in the damped energy norm. Numerical experiments in one dimension illustrate robustness, effectiveness of the estimator, and the accuracy of the reconstructions for standing and propagating waves. The results provide a practical tool for certifying fully discrete leapfrog solutions in wave-dominated simulations, under a fixed time step and mesh, using a damped energy framework.

Abstract

We derive a posteriori error estimates for the the scalar wave equation discretized in space by continuous finite elements and in time by the explicit leapfrog scheme. Our analysis combines the idea of invoking extra time-regularity for the right-hand side, as previously introduced in the space semi-discrete setting, with a novel, piecewise quartic, globally twice-differentiable time-reconstruction of the fully discrete solution. Our main results show that the proposed estimator is reliable and efficient in a damped energy norm. These properties are illustrated in a series of numerical examples.
Paper Structure (23 sections, 13 theorems, 139 equations, 2 figures)

This paper contains 23 sections, 13 theorems, 139 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Continuous and discrete settings
  3. Time and frequency domains
  4. A priori estimates on the exact solution
  5. Discrete problem
  6. CFL condition
  7. Approximation factor
  8. Time reconstructions for the leapfrog scheme
  9. Definitions and key properties
  10. Rewriting of the leapfrog scheme
  11. Stability and approximation properties
  12. A priori estimates on the time-reconstruction error
  13. Asymptotically constant-free error upper bound
  14. Error lower bound
  15. Numerical results
  16. ...and 8 more sections

Key Result

Lemma 2.1

Assume that $u \in C^1_{\rm b}(J;L^2(\Omega))\cap C^0_{\rm b}(J;V)$. The following holds:

Figures (2)

  • Figure 1: Time profiles in the standing wave example
  • Figure 2: Propagating wave: space profiles of the solution at various times

Theorems & Definitions (34)

  • Lemma 2.1: A priori estimate
  • proof
  • Remark 2.2: Bound on undamped energy
  • Remark 2.3: Time-horizon
  • Lemma 2.4: Bound on approximation factor
  • proof
  • Lemma 3.1: Time-reconstruction
  • proof
  • Lemma 3.2: Smooth time-reconstruction
  • proof
  • ...and 24 more