# Goal Oriented Time Adaptivity Using Local Error Estimates
## 3 Time Adaptivity Based on Local Error Estimates
### Convergence in the Solution  
We first relate global errors to variable, non-adaptive timesteps and then consider the adaptive case. The basis is the following Lemma ([25], p. 68):  
**Lemma 1**. : _Consider Problem (1) and a scheme \(\mathcal{N}^{t,\Delta t}\) of order \(p\). Assume a grid \(t_{0}<\cdots<t_{e}\) with timesteps \(\Delta t_{n}=t_{n+1}-t_{n}\) given by a step-size function \(\theta:[t_{0},t_{e}]\to[\theta_{min},1]\), \(\theta_{min}>0\) such that for a given base step size \(\Delta T\)_  
\[\Delta t_{n}=\theta(t_{n})\Delta T.\]_Then, the global error (Equation (7)) fulfills \(\mathbf{e}_{n}=\mathcal{O}(\Delta T^{p})\)._  
Assuming \(\phi_{-}^{\min}:=\min_{t\in[t_{0},t_{c}]}\|\mathbf{\phi}_{-}(t,\mathbf{u}(t))\|>0\) we can define reference timesteps \(\Delta t_{n}^{\text{ref}}\) by a step-size function \(\theta^{\text{ref}}\) and a \(\Delta T_{\text{ref}}\) as follows  
\[\theta^{\text{ref}}(t_{n}):=\left(\frac{\phi_{-}^{\min}}{\|\mathbf{ \phi}_{-}(t_{n-1},\mathbf{u}(t_{n-1}))\|}\right)^{1/(\hat{p}+1)}, \theta^{\text{ref}}(0)=\left(\frac{\tau}{c_{0}^{\text{ref}}} \right)^{1/(\hat{p}+1)},\,(c_{0}^{\text{ref}}>0),\] \[\Delta T_{\text{ref}}=\left(\frac{\tau}{\min\{c_{0}^{\text{ref}},\phi_{-}^{\min}\}}\right)^{1/(\hat{p}+1)}.\]  
These variable, non-adaptive timesteps meet the requirements of Lemma 1, yielding \(\mathbf{e}_{n}=\mathcal{O}(\Delta T_{\text{ref}}^{p})=\mathcal{O}(\tau^{p/(\hat{p }+1)})\). Now, one can observe that Lemma 1 still holds for a grid perturbed by  
\[\Delta t_{n}=\theta(t_{n})\Delta T+\mathcal{O}(\Delta T^{2}), \tag{11}\]  
see [23]. This is straight forward to show using the scaling ansatz \(\theta^{*}(t_{n}):=\theta(t_{n})/c+\mathcal{O}(\Delta T)\) with \(\Delta T^{*}:=c\,\Delta T\). To prove convergence for the adaptive method with the controller in Equation (10), it thus suffices to show that these adaptive steps are a sufficiently small perturbation of the reference steps \(\Delta t_{n}^{\text{ref}}\). **Theorem 1**. : _Consider Problem (1) and an adaptive method consisting of: A pair of schemes \(\mathcal{N}_{+}^{t,\Delta t}\), \(\mathcal{N}_{-}^{t,\Delta t}\) with orders \(p\), \(\hat{p}\), with \(p>\hat{p}\), error estimator Equation (8), deadbeat controller Equation (10), and \(\Delta t_{0}=(\tau/c_{0})^{1/(\hat{p}+1)}+\mathcal{O}(\tau^{2/(\hat{p}+1)})\), \(c_{0},\Delta t_{0}>0\). If the principal error function \(\mathbf{\phi}_{-}\) to \(\mathcal{N}_{-}^{t,\Delta t}\) fulfills_  
\[\min_{t\in[t_{0},t_{c}]}||\mathbf{\phi}_{-}(t,\mathbf{u}(t))||>0, \tag{12}\]  
_then the adaptive method is convergent with \(\mathbf{e}_{n}=\mathcal{O}(\tau^{p/(\hat{p}+1)})\)._  
Proof. : We show by induction over \(n\) that the adaptive timesteps fulfill  
\[\Delta t_{n}=\Delta t_{n}^{\text{ref}}+\mathcal{O}(\Delta T_{\text{ref}}^{2}) \quad\text{implying}\quad\Delta t_{n}=\mathcal{O}(\tau^{1/(\hat{p}+1)}), \tag{13}\]  
by definition of \(\Delta t_{n}^{\text{ref}}\). Choosing \(c_{0}^{\text{ref}}=c_{0}\) for \(\Delta t_{0}^{\text{ref}}\), the _induction base_ is met. The timesteps given by the controller in Equation (10) are  
\[\Delta t_{n+1}=\Delta t_{n}\left(\frac{\tau}{\|\hat{\mathbf{\ell}}_{n}\|}\right)^ {1/(\hat{p}+1)}=\left(\frac{\tau}{\|\mathbf{\phi}_{-}(t_{n},\mathbf{u}_{n})+\mathcal{O }(\Delta t_{n})\|}\right)^{1/(\hat{p}+1)}. \tag{14}\]  
In the denominator we expand \(\mathbf{\phi}_{-}\) in \((t_{n},\mathbf{u}(t_{n}))\), which gives an \(\mathcal{O}(\mathbf{e}_{n})\) term. Since \(\mathbf{e}_{n}\) is based on the steps \(\Delta t_{0}\), \(\ldots,\Delta t_{n-1}\), which fulfill Equation (13), we get \(\mathcal{O}(\mathbf{e}_{n})=\mathcal{O}(\tau^{p/(\hat{p}+1)})\) from Lemma 1. This term is smaller than the \(\mathcal{O}(\Delta t_{n})\) term already in the denominator in Equation (14).