Finite element error analysis of wave equations with dynamic boundary conditions: L2 estimates

Abstract

Abstract $L^2$ norm error estimates of semi- and full discretizations of wave equations with dynamic boundary conditions, using bulk–surface finite elements and Runge–Kutta methods, are studied. The analysis rests 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 2. These can also be observed in the presented numerical experiments.