Numerical Analysis of Nonlinear Partial Differential-Algebraic Equations: A Coupled and an Abstract Systems Approach

Various mathematical models in many application areas give rise to systems of so called partial or abstract differential-algebraic equations (ADAEs). A substantial mathematical treatment of nonlinear ADAEs is still at an initial stage.In this thesis two approaches for treating nonlinear ADAEs are presented. The first one represents an extension of an approach by Tischendorf for the treatment of a specific class of linear ADAEs to the nonlinear case. It is based on the Galerkin approach and the theory of monotone operators for evolution equations. Unique solvability of the ADAE and strong convergence of the Galerkin solutions is proven. Furthermore it is shown that this class of ADAEs has Perturbation Index 1 and at most ADAE Index 1. In the second approach we formulate two prototypes of coupled systems where a semi-explicit differential-algebraic equation is coupled to an infinite dimensional algebraic operator equation or an evolution equation. For both prototypes unique solvability, strong convergence of Galerkin solutions and a Perturbation Index 1 result is shown. Both prototypes can be applied to concrete coupled systems in circuit simulation relying on a new global solvability result for the nonlinear equations of the Modified Nodal Analysis under suitable topological assumptions.