Open Access

Christoph Zimmer

• This thesis is devoted to the application and analysis of time integration schemes for differential-algebraic equations (DAEs) stated in (abstract) Banach spaces. The existence, uniqueness, and regularity of solutions of these so-called operator DAEs are analyzed with the help of temporal discretization methods. The convergence behavior of the time-discrete approximations and their convergence orders are addressed as well. Besides being of interest as a generalization of the concept of DAEs to the infinite-dimensional setting, operator DAEs are an abstract approach for the analysis of constrained partial differential equations (PDEs) in their weak form. The constraints on the solution of the PDEs are possibly given by spatial differential operators like the divergence-free condition on the velocity field in the incompressible Navier-Stokes equations. Examples of constrained PDEs appear in all kinds of physical fields such as fluid dynamics, thermodynamics, electrodynamics,This thesis is devoted to the application and analysis of time integration schemes for differential-algebraic equations (DAEs) stated in (abstract) Banach spaces. The existence, uniqueness, and regularity of solutions of these so-called operator DAEs are analyzed with the help of temporal discretization methods. The convergence behavior of the time-discrete approximations and their convergence orders are addressed as well. Besides being of interest as a generalization of the concept of DAEs to the infinite-dimensional setting, operator DAEs are an abstract approach for the analysis of constrained partial differential equations (PDEs) in their weak form. The constraints on the solution of the PDEs are possibly given by spatial differential operators like the divergence-free condition on the velocity field in the incompressible Navier-Stokes equations. Examples of constrained PDEs appear in all kinds of physical fields such as fluid dynamics, thermodynamics, electrodynamics, mechanics, chemical kinetics, as well as in multi-physical applications where different physical domains are coupled. The first main results of this thesis cover the existence, uniqueness, and regularity of solutions of semi-linear, semi-explicit operator DAEs. In this analysis, the challenges known for DAEs and PDEs have to be tackled simultaneously. These include a limited set of feasible initial values, requirements on the temporal and spatial regularity of the data, and a high sensitivity to perturbations. For operator DAEs with time-independent operators, continuity results for the solutions in the data are used to extend well-known existence, uniqueness, and regularity results to systems with less regular or state-dependent right-hand sides. Similar results for operator DAEs with time-dependent operators are derived by studying the convergence of time-discrete solutions obtained by the implicit Euler method. In this study, time-varying inner products as well as time-dependent kernels of the constraints operators complicate the analysis. As the second main topic, the convergence of the temporal discretization of semi-explicit operator DAEs by implicit, algebraically stable Runge-Kutta methods and explicit exponential integrators is analyzed. As expected from the theory of DAEs and PDEs, the convergence properties depend strongly on the assumed temporal and spatial regularity of the data, vary for the single variables, and differ from finite-dimensional systems. For Runge-Kutta schemes, a regularization is introduced and the strong convergence of the time-discrete approximations under minimal regularity assumptions is proven. A convergence order of q + 1 and of q + 1/2 is shown for the state and the Lagrange multiplier, respectively. Here, q denotes the stage order of the Runge-Kutta scheme. For explicit exponential integrators, order conditions for methods up to order three are derived for the state of semi-linear operator DAEs. In addition, an approximation of the Lagrange multiplier is introduced whose convergence order is reduced by half an order. For both classes of integration schemes, sufficient conditions are formulated which increase the convergence order. The results are supported by numerical examples.…