A Posteriori Single- and Multi-Goal Error Control and Adaptivity for Partial Differential Equations
<|reference_start|>A Posteriori Single- and Multi-Goal Error Control and Adaptivity for Partial Differential Equations: 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.<|reference_end|>
@article{endtmayer2024a,
    title={A Posteriori Single- and Multi-Goal Error Control and Adaptivity for
  Partial Differential Equations},
    author={Bernhard Endtmayer, Ulrich Langer, Thomas Richter, Andreas Schafelner
  and Thomas Wick},
    journal={arXiv preprint arXiv:2404.01738},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.01738},
    primaryClass={math.NA cs.NA}
}