A reliable direct numerical treatment of differential–algebraic equations by overdetermined collocation: An operator approach

Recently reported experiments and theoretical contributions concerning overdetermined polynomial collocation applied to higher-index differential–algebraic equations give rise to the conjecture that next to the existing derivative-array based methods there is further potential toward a reliable direct numerical treatment of DAEs. By analyzing first-order differential–algebraic operators and their special approximations in detail, we contribute to justify the overdetermined polynomial collocation applied to first-order higher-index differential–algebraic equations and fill the hitherto existing gap between the theoretical convergence results and its practical realization. Moreover, we shortly touch related questions for higher-order DAEs. We discuss several practical aspects of higher-order differential–algebraic operators and the associated equations which may be important for the application of collocation methods.