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

Bernhard Endtmayer, Ulrich Langer, Thomas Richter, Andreas Schafelner, Thomas Wick

TL;DR

The paper develops and surveys a posteriori, goal-oriented error estimation for PDEs using the dual-weighted residual method with partition-of-unity localization, extending from single- to multi-goal functionals. It introduces enriched- and interpolated-space estimators to obtain computable, efficient, and reliable error indicators, along with strategies to balance discretization and nonlinear iteration errors. A key theoretical advance is a saturation-based efficiency/reliability framework that requires only one saturation assumption, plus detailed treatment of non-standard discretizations and non-conforming methods. The approach is validated through Poisson, nonlinear elliptic, stationary Navier–Stokes, and space-time parabolic p-Laplace applications, with open-source implementations to support reproduction and further development. The results underscore the practical impact of space-time multigoal adaptivity for complex multiphysics problems and guide future work on extending error estimators to additional error sources and model-order reductions.

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.

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

TL;DR

The paper develops and surveys a posteriori, goal-oriented error estimation for PDEs using the dual-weighted residual method with partition-of-unity localization, extending from single- to multi-goal functionals. It introduces enriched- and interpolated-space estimators to obtain computable, efficient, and reliable error indicators, along with strategies to balance discretization and nonlinear iteration errors. A key theoretical advance is a saturation-based efficiency/reliability framework that requires only one saturation assumption, plus detailed treatment of non-standard discretizations and non-conforming methods. The approach is validated through Poisson, nonlinear elliptic, stationary Navier–Stokes, and space-time parabolic p-Laplace applications, with open-source implementations to support reproduction and further development. The results underscore the practical impact of space-time multigoal adaptivity for complex multiphysics problems and guide future work on extending error estimators to additional error sources and model-order reductions.

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 -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.
Paper Structure (81 sections, 17 theorems, 185 equations, 31 figures, 10 tables, 3 algorithms)

This paper contains 81 sections, 17 theorems, 185 equations, 31 figures, 10 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notation, Abstract Setting, Finite Element Discretization
  3. Abstract Setting
  4. Finite Element Discretization
  5. Finite Elements on Simplices $P_k$
  6. Finite Elements on Hypercubes $Q_k$
  7. Single-Goal Oriented Error Control
  8. Motivation and Preliminaries
  9. Adjoint Problem
  10. An Error Identity
  11. Error Estimation in Enriched Spaces
  12. Parts of the Error Estimator for Enriched Spaces
  13. The Remainder Part $\eta_{\mathcal{R}}^{(2)}$ or Linearization Error Estimator.
  14. The Iteration Error Estimator $\eta_k$
  15. The Discretization Error Estimator $\eta_h^{(2)}$
  16. ...and 66 more sections

Key Result

Theorem 1

Let $\tilde{u} \in U$ and $\tilde{z} \in V$ be arbitrary but fixed, and let $u \in U$ be the solution of the model problem eq: General Modelproblem, and $z \in V$ be the solution of the adjoint problem eq: General Adjointproblem. If $\mathcal{A} \in \mathcal{C}^3(U,V^*)$ and $J \in \mathcal{C}^3(U,\ for arbitrary but fixed $\tilde{u} \in U$ and $\tilde{z} \in V$, where and the remainder term wit

Figures (31)

  • Figure 1: Visualization of how the degrees of freedom are interpolated on elements of higher order with a coarser grid for $Q_1$ finite elements.
  • Figure 2: Visualization of how the degrees of freedom are interpolated on elements of higher order with a coarser grid for $P_1$ finite elements.
  • Figure 3: Nodal error estimator on the hanging $\eta_5$ is distributed equally to $\eta_3$ and $\eta_5$ the nodal contribution is distributed to the elements.
  • Figure 4: Example 1: Going from left to right and top to bottom: error indicators, adaptive mesh, primal solution and adjoint solution.
  • Figure 5: Example 2: Initial mesh with quantities of interest (left), and adaptively refined mesh after 25 refinement steps driven by the combined functional $J_\mathfrak{E}$ (right).
  • ...and 26 more figures

Theorems & Definitions (43)

  • Theorem 1: see RanVi2013EnLaWi18
  • proof
  • Remark 1
  • Theorem 2: see endtmayer2023goal
  • proof
  • Remark 2
  • Definition 1: Efficient and Reliable
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 33 more