5. Conclusion
In this article, we extended two classical results of goal-oriented error estimation to problems solved by domain decomposition methods. We solved simultaneously both forward and adjoint problems on the same mesh using a block conjugate gradient algorithm and proposed a fully parallel procedure to recover admissible fields and estimate the error on the quantity of interest. Moreover, the error bounds separated the contribution of the discretization from the contribution of the iterative solver which lead to defining of a stopping criterion of the iterative solver that prevented oversolving. We also showed that the method could be easily extended to the case of locally enriched kinematics (using h or p refinements or dedicated enrichments) only constrained to contain the original coarse kinematic on the boundary of subdomains. This property provided a simple parallel way to improve estimators as soon as the singularities (local quantity of interests, crack tips, etc.) were not too close to the interface. The concepts were validated on academic examples using the most basic h-refinement strategy. Of course using ad-hoc enrichment instead of brutal h-refinement should improve the performance of the method. Future work will focus on exploiting the fact that the shape of the interface is preserved by the enrichment to define acceleration strategies, and on defining original strategies to balance the computational load between subdomains for the resolution of the refined problem.
