Finite element error analysis of wave equations with dynamic boundary conditions: $L^2$ estimates

TL;DR

This paper provides $L^2$ error estimates for semi- and full discretisations of wave equations with dynamic boundary conditions, revealing surprising convergence behaviors through energy-based analysis and numerical validation.

Contribution

It introduces an abstract framework for error analysis of wave equations with dynamic boundary conditions and uncovers unexpected convergence rates for certain problem types.

Findings

Spatial convergence order is less than two for problems with velocity or acoustic boundary conditions.

Energy techniques are used to derive error estimates within an abstract setting.

Numerical experiments confirm the theoretical error estimates and convergence behaviors.

Abstract

$L^2$ norm error estimates of semi- and full discretisations, using bulk--surface finite elements and Runge--Kutta methods, of wave equations with dynamic boundary conditions are studied. The analysis resides on an abstract formulation and error estimates, via energy techniques, within this abstract setting. Four prototypical linear wave equations with dynamic boundary conditions are analysed which fit into the abstract framework. For problems with velocity terms, or with acoustic boundary conditions we prove surprising results: for such problems the spatial convergence order is shown to be less than two. These can also be observed in the presented numerical experiments.