A Posteriori Single- and Multi-Goal Error Control and Adaptivity for Partial Differential Equations

TL;DR

This paper reviews goal-oriented a posteriori error control and adaptivity techniques for finite element methods applied to various PDEs, emphasizing multi-goal strategies and their theoretical and practical implementations.

Contribution

It extends theoretical results for error efficiency and reliability and demonstrates adaptive algorithms for multiple PDE applications using different finite element discretizations.

Findings

Adaptive algorithms effectively balance discretization and non-linear errors.

Multi-goal error control accurately evaluates multiple quantities of interest.

Open-source implementations facilitate practical application and further research.

Abstract

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.