PDAE Analysis for Coupled Circuit Device Simulation with Finite and Mixed-Finite Elements

Sascha Baumanns , Lennart Jansen , Monica Selva Soto , Caren Tischendorf

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976),

MSC 2000

65M20 Method of lines

34A09 Implicit equations, differential-algebraic equations

Abstract
In the last years several mathematical models coupling differential-algebraic equations and partial differential equations for describing the behavior of electrical circuits have been proposed in the literature. Most of them investigate the properties of coupled systems that include one dimensional drift-diffusion equations for describing the highly sensitive semi-conducting elements in the circuit. Here, we extend the results to coupled systems with higher dimensional drift-diffusion models which allow a proper modeling of different material regions. For stability reasons, we investigate a monolithic simulation approach and consider two common variants of PDE discretizations for semiconductors in the system: besides a finite element method combined with the Scharfetter-Gummel approach, a mixed finite element discretization. The resulting differential algebraic equations share important properties that allow us to show that their index is always less or equal to two and depends only on the circuit's topology. Additionally, we conclude unique solvability of the associated initial value problems. Finally we show simulation results using a self-developed MATLAB code.