Various mathematical models in many application areas give rise to systems of partial differential equations and differential-algebraic equations (DAEs). These systems are called partial or abstract differential-algebraic equations (ADAEs). Being usually discretized by the method of lines the semi-discretized system yields in general a DAE. A substantial mathematical treatment of nonlinear ADAEs is still at an initial stage.
We present two approaches treating nonlinear ADAEs. We investigate them with regard to the solvability and uniqueness of solutions and the convergence of solutions of semi-dicretized systems to the original solution. Furthermore we study the sensitivity of a solution with regard to perturbations on the right hand side and in the initial value.
The first approach 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. We prove unique solvability of the ADAE and strong convergence of the Galerkin solutions. Furthermore we prove 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, an elliptic and a parabolic one. Here a semi-explicit DAE is coupled to an infinite dimensional algebraic operator equation or an evolution equation. For both prototypes we prove unique solvability, strong convergence of Galerkin solutions and a Perturbation Index 1 result. Both prototypes are applied to concrete coupled systems in circuit simulation. In this context we also prove a global solvability result for the nonlinear equations of the Modified Nodal Analysis under suitable topological assumptions
