# Goal Oriented Time Adaptivity Using Local Error Estimates
## 3 Time Adaptivity Based on Local Error Estimates
### The Zero Set of the Density Function and Global Error Propagation  
Now, we analyze global error dynamics to make a qualitative statement about the obtained grid. We establish guidelines to predict grid quality and thus the performance of the goal-oriented adaptive method. Assume a QoI with a density function that is linear in \(\mathbf{u}\),  
\[j(t,\mathbf{u}(t))=\mathbf{w}^{\mathsf{T}}\mathbf{u}(t),\quad w_{i}\geq 0.\]  
Global error propagation only appears in the time-integration part of Equation (17), given by  
\[|\mathcal{J}_{h}(\mathbf{u})-\mathcal{J}_{h}(\mathbf{u}_{h})|\leq\sum_{n=0}^{N-1} \left|\Delta t_{n}\sum_{k=0}^{s}\sigma_{k}\mathbf{w}^{\mathsf{T}}\mathbf{e}_{n}^{(k)} \right|.\]We define a weighted seminorm \(\|\cdot\|_{w}:\mathbb{R}^{d}\mapsto\mathbb{R}\) by  
\[\|\mathbf{x}\|_{w}:=\sum_{i=1}^{d}w_{i}|x_{i}|,\quad w_{i}\geq 0, \tag{30}\]  
for which we assume \(w_{i}=0\) for some indices \(i\), yielding a non-trivial zero set. Using Equation (30) we get  
\[|\mathcal{J}_{h}(\mathbf{u})-\mathcal{J}_{h}(\mathbf{u}_{h})|\leq\sum_{n=0}^{N-1} \Delta t_{n}\sum_{k=0}^{s}\sigma_{k}\|\mathbf{e}_{n}^{(k)}\|_{w}\]  
and are thus interested in \(\|\mathbf{e}_{n}^{(k)}\|_{w}\), resp. \(\|\mathbf{e}_{n}\|_{w}\). The global error propagation form for this is  
\[\|\mathbf{e}_{n+1}\|_{w}\leq\underbrace{\left\|\mathcal{M}^{t_{n}\Delta t_{n}}\bm {e}_{n}\right\|_{w}}_{\text{global error propagation}}+\underbrace{\left\| \left(\mathcal{N}^{t_{n}\Delta t_{n}}-\mathcal{M}^{t_{n}\Delta t_{n}}\right) \mathbf{u}_{n}\right\|_{w}}_{\text{global error increment}}, \tag{31}\]  
compare Equations (19) and (20). Using this, we now consider grid efficiency (number of step vs. global error for a fixed error or number of steps) and how global error dynamics affect the error in the QoI. Assume the local error estimate \(\hat{\mathbf{\ell}}_{n}\), see Equation (8), to be of comparable size in both \(\|\cdot\|_{U}\) and \(\|\cdot\|_{w}\), i.e., \(\|\hat{\mathbf{\ell}}_{n}\|_{w}\approx\|\hat{\mathbf{\ell}}_{n}\|_{U}\). This means both the classical (Equation (10)) and the goal oriented controller in Equation (26), neglecting the quadrature estimate (cf. Remark 3), yield similar step sizes. The question of grid efficiency becomes one of global error dynamics. Now, if we also have \(\|\mathcal{M}\mathbf{e}_{n}\|_{U}\approx\|\mathcal{M}\mathbf{e}_{n}\|_{w}\), the resulting global errors and grid efficiency are similar. Examples are Qols capturing the main dynamics, and thus largest errors, of a system. On the other hand, assume \(\|\mathcal{M}\mathbf{e}_{n}\|_{U}>\|\mathcal{M}\mathbf{e}_{n}\|_{w}\). This results in a smaller error in the QoI and a higher grid efficiency. Examples for this are convection dominated PDEs where \(\mathcal{M}\) shifts global errors to the zero set of \(\|\cdot\|_{w}\). In the following, we consider only the errors \(\mathbf{e}_{n}\) obtained by the goal oriented method. We instead assume \(\|\hat{\mathbf{\ell}}_{n}\|_{w}<\|\hat{\mathbf{\ell}}_{n}\|_{U}\), i.e., parts of the local error estimate to be in the zero set of \(\|\cdot\|_{w}\), resulting in larger timesteps for the goal oriented method. If \(\|\mathbf{e}_{n}\|_{w}\) remains small despite larger timesteps compared to the classical method, grid efficiency is better. Examples are flows \(\mathcal{M}\) such that \(\mathbf{e}_{n}\), despite an increasing norm, largely remains in the zero set of \(\|\cdot\|_{w}\) or when the main dynamics of a system have a negligible effect on the error in the QoI. Another example is a flow \(\mathcal{M}\) such that \(\|\mathbf{e}_{n}\|_{U}\) remains small in the first place due to, for example, strong dampening. An example with a low grid efficiency is obtained if global errors grow in the zero set of \(\|\cdot\|_{w}\) and are later shifted into its image. This can occur if the timesteps of the goal oriented method leaves dynamics in the zero set of \(\|\cdot\|_{w}\) insufficiently resolved, leading to \(\mathbf{e}_{n}\) large in norm. Such error shifts can be caused, for example, by convection in PDEs.