# Goal Oriented Time Adaptivity Using Local Error Estimates
## 3 Time Adaptivity Based on Local Error Estimates
### Convergence in the QoI  
We show convergence in the QoI for the adaptive timesteps (Equation (26)) following the structure of Section 3.2 for showing convergence in the solution for the timesteps of Equation (10). With  
\[\psi(t):=\left|\frac{\partial j(t,\mathbf{u}(t))}{\partial\mathbf{u}}\,\mathbf{\phi}_{-}(t,\mathbf{u}(t))\right|+\left|\mathbf{\phi}_{-}(t,\mathbf{u}(t))\right|,\quad\psi_{\min}:= \min_{t\in[t_{0},t_{c}]}\psi(t)>0, \tag{27}\]  
we define non-adaptive reference timesteps \(\Delta t_{n}^{\text{ref}}=\theta^{\text{ref}}(t_{n})\Delta T_{\text{ref}}\) as follows:  
\[\theta^{\text{ref}}(t_{n}):=\left(\tfrac{\tau_{\min}}{\theta(t_{n-1})}\right) ^{1/(p+1)},\quad\theta^{\text{ref}}(0):=\left(\tfrac{\tau}{\alpha_{0}^{\text{ ref}}}\right)^{1/(p+1)},\,(c_{0}^{\text{ref}}>0),\quad\Delta T_{\text{ref}}:= \left(\tfrac{\tau}{\min\{c_{0},\theta_{\min}\}}\right)^{1/(p+1)}.\]  
We now show that our adaptive timesteps (Equation (26)) are a sufficiently small perturbation (Equation (11)) of the above reference steps. This yields convergence in the solution by Lemma 1 and then convergence in the QoI by Theorem 2. **Theorem 3**. : _Consider Problem (1) and (2) and assume \(j\) to be sufficiently smooth. Assume an adaptive method consisting of: Time-integration schemes \(\mathcal{N}_{+}^{t,\Delta t}\), \(\mathcal{N}_{-}^{t,\Delta t}\) with orders \(p\), \(\hat{p}\), \(p>\hat{p}\), quadrature schemes of order \(r\), \(\hat{r}=\hat{p}\), and \(r>\hat{r}\), a scheme \(\hat{\mathcal{N}}^{t,\Delta t}\) to obtain solutions of order \(p-1\) for intermediate points, the error estimator in Equation (21), controller in Equation (26), and \(\Delta t_{0}=(\tau/c_{0})^{1/(p+1)}+\mathcal{O}(\tau^{2/(p+1)})\), \(c_{0},\Delta t_{0}>0\). If the principle error functions corresponding to the lower order schemes fulfill Equation (27), then \(e^{\mathcal{J}}=|\mathcal{J}(\mathbf{u})-\mathcal{J}_{h}(\mathbf{u}_{h})|=\mathcal{O}( \tau^{\min(r,\,p)/(p+1)})\)._  
Proof. : We first show convergence in the solution by showing  
\[\Delta t_{n}=\Delta t_{n}^{\text{ref}}+\mathcal{O}(\Delta T_{\text{ref}}^{2}), \tag{28}\]  
via induction over \(n\). The induction base is met by choosing \(c_{0}^{\text{ref}}=c_{0}\) for \(\Delta t_{0}^{\text{ref}}\). For the steps in Equation (26), we address the denominator terms separately. For the local error estimate we have  
\[\hat{\ell}_{n}^{\mathcal{J}_{n}}=j\left(t_{n+1},\mathcal{N}_{-}^{t_{n},\Delta t _{n}}\mathbf{u}_{n}\right)-j\left(t_{n+1},\mathcal{N}_{+}^{t_{n},\Delta t_{n}}\bm {u}_{n}\right)=\frac{\partial j(t_{n+1},\mathbf{u}_{n+1})}{\partial\mathbf{u}}\hat{ \ell}_{n}+\mathcal{O}(\Delta t_{n}^{2\hat{p}+2}).\]  
Using the same expansions as in the proof for Theorem 1 we have  
\[\hat{\mathbf{\ell}}_{n}=\Delta t_{n}^{\hat{p}+1}\left(\mathbf{\phi}_{-}(t_{n},\mathbf{u}(t _{n}))+\mathcal{O}(\Delta t_{n})\right).\]  
Similarly, we get  
\[\frac{\partial j(t_{n+1},\mathbf{u}_{n+1})}{\partial\mathbf{u}}=\frac{\partial j(t_{n},\mathbf{u}(t_{n}))}{\partial\mathbf{u}}+\mathcal{O}(\mathbf{e}_{n})+\mathcal{O}(\Delta t_ {n}).\]  
The steps \(\Delta t_{0}\), \(\ldots,\Delta t_{n-1}\) leading up to \(t_{n}\) fulfill the induction hypothesis in Equation (28) and thus are a perturbation of \(\Delta t_{i}^{\text{ref}}\) fulfilling Equation (11). Thus, Lemma 1 holds, which gives \(\mathcal{O}(\mathbf{e}_{n})=\mathcal{O}(\Delta T_{\text{ref}}^{p})\). Thus, we obtain  
\[\hat{\ell}_{n}^{\mathcal{J}_{n}}=\Delta t_{n}^{\hat{p}+1}\left(\frac{\partial j (t_{n},\mathbf{u}(t_{n}))}{\partial\mathbf{u}}\mathbf{\phi}_{-}(t_{n},\mathbf{u}(t_{n}))+ \mathcal{O}(\Delta t_{n})+\mathcal{O}(\Delta T_{\text{ref}}^{p})\right). \tag{29}\]For the quadrature estimate we use Equation (24).