# Goal Oriented Time Adaptivity Using Local Error Estimates
## 1 Introduction
We aim to extend these methods to problems like in Equations (1) and (2). One way to include the QoI [13] is to reformulate Equations (1) and (2) as  
\[\frac{d}{dt} \begin{bmatrix}\mathbf{u}(t)\\ I(t)\end{bmatrix} = \begin{bmatrix}f(t,\mathbf{u}(t))\\ j(t,\mathbf{u}(t))\end{bmatrix} ,\quad t\in[t_{0},t_{e}],\quad \begin{bmatrix}\mathbf{u} (t_{0})\\ I(t_{0})\end{bmatrix} = \begin{bmatrix}\mathbf{u}_{0}\\ 0\end{bmatrix} . \tag{3}\]  
We then have \(J(t_{e})=\int_{t_{0}}^{t_{e}}j(t,\mathbf{u}(t))dt=\mathcal{J}(\mathbf{u})\). Solving Equation (3) using an adaptive method based on the norm of the local error, the additional term coming from the added equation acts as a quadrature error estimate. In our numerical tests, this additional term is negligible in comparison to the local error in \(\mathbf{u}\), and thus there is no benefit compared to the classic method. Here, \(\mathcal{J}\) is discretized using the quadrature scheme defined by time-integration. This choice has an impact on performance, but we do not have conclusive results when it is beneficial or detrimental. A different approach is to combine the classic adaptive method with an error estimate that uses local errors in the QoI only [14]. This way, the hope is to be more efficient than either DWR or the classic approach by having a cheap error estimate, focusing on direct error contributions to the QoI. One only requires additional evaluations of \(j(t,\mathbf{u}(t))\) to compute the error estimate. This makes it notably less intrusive to implement compared to DWR and suitable for partitioned approaches to coupled systems, since it only requires a local error estimate. Methods in this spirit have been proposed before. [PERSON] and [PERSON] conceptualize this idea for drag and lift coefficients in incompressible flows [15]. [PERSON] predicts \"catastrophical results\" [16] for a problem with an alternating lift coefficient due to vanishing error estimates. As a remedy, [PERSON] proposes the usage of multiple QoIs for error estimation and does so using pressure, velocity, lift and drag. [PERSON] et al. use a point-wise evaluation of the displacement field in fluid-structure interaction [17; 18]. The authors get good results in two test cases, in one of which they observe slightly larger errors for too large tolerances. However, in all of the above cases, usage of this approach is heuristic and no analysis is presented. Thus the question arises if these reports can be explained and could have been expected. The purpose of this article is to present a formal derivation and analysis of such a goal oriented adaptive method based on local error estimates, allowing us to answer this question in detail. To this end, we estimate the contribution per timestep to the error in the QoI. It consists of both quadrature and time-integration errors. We then derive local error estimates for these and use them in a deadbeat controller. For the new adaptive method, we show convergence in the solution under a controllability condition with additional requirements on the timesteps. To obtain tolerance proportionality, i.e., \(|\mathcal{J}(\mathbf{u})-\mathcal{J}_{h}(\mathbf{u}_{h})|=\mathcal{O}(\tau)\) (we define \(\mathbf{e}(t)=\mathcal{O}(t^{\beta})\), iff \(\exists\,C>0,\ \delta>0:\|\mathbf{e}(t)\|\leq Ct^{\beta},\quad\forall\ 0<t\leq\delta\)), for the error in the QoI when using higher order (\(>\)2) time-integration schemes, one needs solutions of sufficiently high order in intermediate points. We do our analysis for embedded RK schemes [11] and a deadbeat timestep controller. The structure of our proof allows for simple convergence proofs of other controllers, for example, PID controllers [19], based on similarity to the deadbeat controller. To make statements on the performance of the goal oriented adaptive method, we analyze the impact of error dynamics on the error in the QoI. We use numerical tests with widely different global error behaviors with respect to the QoI. For these, we confirm the convergence results and are able to explain the performance results.