A unified error analysis for the numerical solution of nonlinear wave-type equations with application to kinetic boundary conditions

In this thesis, a unified error analysis for discretizations of nonlinear first- and second-order wave-type equations is provided. For this, the wave equations as well as their space discretizations are considered as nonlinear evolution equations in Hilbert spaces. The space discretizations are supplemented with Runge-Kutta time discretizations. By employing stability properties of monotone operators, abstract error bounds for the space, time, and full discretizations are derived.

Further, for semilinear second-order wave-type equations, an implicit-explicit time integration scheme is presented. This scheme only requires the solution of a linear system of equations in each time step and it is stable under a step size restriction only depending on the nonlinearity. It is proven that the scheme converges with second order in time and in combination with the abstract space discretization of the unified error analysis, corresponding full discretization error bounds are derived.

The abstract results are used to derive convergence rates for an isoparametric finite element space discretization of a wave equation with kinetic boundary conditions and nonlinear forcing and damping terms. ... mehr
For the combination of the finite element discretization with Runge-Kutta methods or the implicit-explicit scheme, respectively, error bounds of the resulting fully discrete schemes are proven. The theoretical results are illustrated by numerical experiments.