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Finite Element Convergence Analysis For Wave Equations With Time-Dependent Coefficients

Oussama Al Jarroudi, Marcus J. Grote

TL;DR

This work develops a rigorous finite element convergence theory for second-order hyperbolic equations with coefficients that vary in space and time, addressing challenges from time-dependent operators that break energy conservation. By introducing a time-dependent Ritz-like projection and employing a duality argument, the authors prove optimal $H^1$-norm error estimates for semi-discrete FEM solutions. The theory is complemented by numerical experiments in 1D that confirm the predicted convergence rates and reveal localized wave-field amplification in time-modulated metamaterial-like structures. The results extend FEM error analysis to dynamic media, with implications for accurate simulation and control of waves in time-varying materials.

Abstract

Error estimates are proved for finite element approximations to the solution of second-order hyperbolic partial differential equations with coefficients varying in both space and time. Optimal rates of convergence in the energy norm are proved for the semi-discrete Galerkin finite element solution by introducing a time-dependent Ritz-like projection. Numerical experiments corroborate the rates of convergence and illustrate the localized wave field enhancement in a chain of time-modulated subwavelength resonators.

Finite Element Convergence Analysis For Wave Equations With Time-Dependent Coefficients

TL;DR

This work develops a rigorous finite element convergence theory for second-order hyperbolic equations with coefficients that vary in space and time, addressing challenges from time-dependent operators that break energy conservation. By introducing a time-dependent Ritz-like projection and employing a duality argument, the authors prove optimal -norm error estimates for semi-discrete FEM solutions. The theory is complemented by numerical experiments in 1D that confirm the predicted convergence rates and reveal localized wave-field amplification in time-modulated metamaterial-like structures. The results extend FEM error analysis to dynamic media, with implications for accurate simulation and control of waves in time-varying materials.

Abstract

Error estimates are proved for finite element approximations to the solution of second-order hyperbolic partial differential equations with coefficients varying in both space and time. Optimal rates of convergence in the energy norm are proved for the semi-discrete Galerkin finite element solution by introducing a time-dependent Ritz-like projection. Numerical experiments corroborate the rates of convergence and illustrate the localized wave field enhancement in a chain of time-modulated subwavelength resonators.
Paper Structure (9 sections, 3 theorems, 101 equations, 4 figures)

This paper contains 9 sections, 3 theorems, 101 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Wave equation in a time-varying medium
  3. Finite element formulation
  4. A time-dependent projection
  5. Semi-discrete finite-element approximation
  6. Numerical experiments
  7. Time-modulated $\rho(x,t)$ and $\kappa(x,t)$
  8. Time-modulated $\rho(x,t)$, $\kappa(x,t)$ and $\sigma(x,t)$
  9. Chain of time-modulated subwavelength resonators

Key Result

Theorem 4.1

Let $u$ be the solution to eq:wave with $u_t \in L^{\infty}\left(0,T; H^r(\Omega)\right)$ for $r\geq 2$, and $w(t)$ be the time-dependent projection defined in TimeDepProj. Then, there exists a constant $C>0$, independent of $h$, such that it holds for all t $\leqslant T$:

Figures (4)

  • Figure 1: Time-modulated $\rho(x,t)$ and $\kappa(x,t)$: $L^2$-error (blue) and $H^1$-error (red). The dashed lines indicate the expected convergence rates $O(h^2)$ and $O(h^3)$ for comparison.
  • Figure 2: Time-modulated $\rho(x,t)$, $\kappa(x,t)$, and $\sigma(x,t)$: $L^2$-error (blue) and $H^1$-error (red). The dashed lines indicate the expected convergence rates $O(h^2)$ and $O(h^3)$ for comparison.
  • Figure 3: Schematic of three time-modulated resonators $R_i$ (green) embedded in a background medium (dashed blue line). The distance between consecutive resonators is denoted by $\ell_{i,i+1}$.
  • Figure 4: Chain of time-modulated subwavelength resonators: snapshots of the numerical solution at times $t = 238, 500, 1164, 1236$.

Theorems & Definitions (5)

  • Theorem 4.1
  • Proposition 4.2
  • proof
  • Theorem 5.1
  • proof