# Goal Oriented Time Adaptivity Using Local Error Estimates
## 6 Conclusions  
Similar to previously proposed methods, we derived a simple and easy to implement goal oriented local error estimator. Implementation comes with little to no extra work and is well-suited for partitioned approaches for coupled systems.  
For the resulting goal oriented adaptive method, we show convergence for the first time and its rates with respect to the error in the QoI. To obtain tolerance proportionality, sufficiently high order solutions in all quadrature evaluation points are needed. The proof is constructive and gives rise to a principle to formally show convergence for closely related stepsize controllers.  
Performance of the goal oriented method deteriorates when local errors of underresolved processes accumulate in the zero set of the density function that later shift into its image. This can be the case for convection dominated problems, and one should then consider using the standard local error estimate. It is less problematic for dissipative problems, where slow and global processes dominate the error.  
Numerical experiments designed to test these guidelines show them to work well, as well as confirming the results on convergence rates. The tests show that bad performance of the goal oriented  
Figure 6: Comparison of the local error based adaptive methods for the coupled heat problem.  
method can be predicted and thus avoided. For dissipative test cases, the DWR method is notably more expensive and outperformed by the local error based method. The goal oriented adaptive method shows good performance in most cases and significant speedups in some.  
Future WorkIt would be interesting to test the proposed goal oriented method in the context of PDE-constrained optimization. There, one has to repeatedly solve a given PDE for varying parameters. This would be particularly interesting for non-gradient based optimization algorithms, where the proposed method could lead to considerable speedups for the optimization process as a whole.  
Our analysis is based on a specific type of QoI. It is natural to try to extend our results to more general QoIs, for example, point-wise evaluations in time.  
One could further generalize this work by using a density function \(j_{\text{est}}\) in error estimation, that is not the density function \(j\) from the QoI. This adds a degree of freedom in constructing the goal oriented method and can be used to overcome possible weaknesses, such as vanishing error estimates. This requires further testing and gives rise to the question on how to optimally choose \(j_{\text{est}}\).  
P.M.: investigation, methodology, software, validation, formal analysis, visualization, writing. P.B.: conceptualization, funding acquisition, methodology, formal analysis, supervision, writing. All authors have read and agreed to the published version of the manuscript.  
The authors gratefully acknowledge support from the Crafoord Foundation under grant number 20150681.  
The authors want to thank [PERSON] for helping with FEniCS and dolfin-adjoint, [PERSON] and [PERSON] for valuable feedback for many interesting discussions, and [PERSON] for the implementation of the final test problem.  
The authors declare no conflict of interest.