This work reviews goal-oriented a posteriori error control, adaptivity and solver control for finite element approximations to boundary and initial-boundary value problems for stationary and non-stationary partial differential equations, respectively. In particular, coupled field problems with different physics may require simultaneously the accurate evaluation of several quantities of interest, which is achieved with multi-goal oriented error control. Sensitivity measures are obtained by solving an adjoint problem. Error localization is achieved with the help of a partition-of-unity. We also review and extend theoretical results for efficiency and reliability by employing a saturation assumption. The resulting adaptive algorithms allow to balance discretization and non-linear iteration errors, and are demonstrated for four applications: Poisson's problem, non-linear elliptic boundary value problems, stationary incompressible Navier-Stokes equations, and regularized parabolic $p$-Laplace initial-boundary value problems. Therein, different finite element discretizations in two different software libraries are utilized, which are partially accompanied with open-source implementations on GitHub.

翻译：本文综述了面向偏微分方程的稳态与非稳态边值问题和初边值问题的有限元近似中的目标导向后验误差控制、自适应及求解器控制。特别地，涉及不同物理机制的耦合场问题可能需要同时精确评估多个关注量，这可通过多目标导向误差控制实现。灵敏度度量通过求解伴随问题获得，误差定位则借助单位分解技术完成。我们还回顾并拓展了基于饱和假设的效率和可靠性理论结果。由此产生的自适应算法能够平衡离散化误差与非线性迭代误差，并通过四个应用实例进行验证：泊松问题、非线性椭圆边值问题、稳态不可压缩纳维-斯托克斯方程以及正则化抛物型$p$-拉普拉斯初边值问题。其中，采用两个不同软件库中的多种有限元离散化方法，部分实现代码已在GitHub上以开源形式发布。