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

TL;DR

This paper establishes error estimates and convergence rates for finite element methods applied to wave equations with coefficients that change over space and time, supported by numerical experiments.

Contribution

It introduces a novel time-dependent Ritz-like projection to prove optimal convergence rates for hyperbolic PDEs with variable coefficients.

Findings

Optimal convergence rates in energy norm are achieved.

Numerical experiments confirm theoretical error estimates.

Wave field enhancement in time-modulated resonators is demonstrated.

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.