Adjoint-based A Posteriori Error Analysis for Semi-explicit Index-1 and Hessenberg Index-2 Differential-Algebraic Equations

In this work we develop adjoint-based analyses for $\mathit{\text{a}}\mathit{\text{posteriori}}$ error estimation for the temporal discretization of differential-algebraic equations (DAEs) of special type: semi-explicit index-1 and Hessenberg index-2. Our technique quantifies the error in a Quantity of Interest (QoI), which is defined as a bounded linear functional of the solution of a DAE. We derive representations for errors of various types of QoIs (depending on the entire time interval, final time, algebraic variables, differential variables, etc.). We develop two analyses: one that defines the adjoint to the DAE system, and one that first converts the DAE to an ODE system and then applies classical $\mathit{\text{a}}\mathit{\text{posteriori}}$ analysis techniques. A number of examples are presented, including nonlinear and non-autonomous DAEs, as well as spatially discretized partial differential-algebraic equations (PDAEs). Numerical results indicate a high degree of accuracy in the error estimation.