Computational methods for wave-type problems with non-standard boundary conditions

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This project is devoted to the numerical approximation and numerical analysis of initial-boundary value problems of wave type with kinetic and acoustic boundary conditions. Such non-standard boundary conditions are indispensable if the effective properties on the surface need to be reflected properly. A representative example is the modeling of membrane vibrations of a bass drum, where standard Dirichlet or Neumann boundary conditions would fail. In situations of strong nonlinearities or heterogeneities on the boundary, which then lead to different characteristic length scales in the bulk and on the boundary, the numerical simulation of such problems depicts a challenging task. Within this project, we consider an alternative problem formulation as a partial differential-algebraic equation, which allows a tailored numerical treatment. In particular, it is possible to apply different mesh sizes or even different discretization schemes in the bulk and on the surface. This approach yields a novel class of approximation schemes for wave-type systems with non-standard boundary conditions. The corresponding analysis combines mixed finite element schemes with techniques known from the field of differential-algebraic equations. Moreover, the project aims to generalize Gautschi-type wave integrators to constrained systems. This would allow an efficient integration of the proposed alternative model formulation by an explicit time stepping scheme without a restrictive step size restriction. Another major part of this project deals with the design and analysis of bulk-surface splitting schemes. Therein, the idea is to solve the bulk and surface dynamics in an alternating manner. Hence, the system is decoupled and enables more flexible and tailored approximation schemes in space and in time. All together, this then yields efficient simulation tools for wave-type systems with non-standard boundary conditions.