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A short perspective on a posteriori error control and adaptive discretizations

Roland Becker, Stéphane P. A. Bordas, Franz Chouly, Pascal Omnes

TL;DR

The work surveys a posteriori error estimation and adaptive discretization as a framework to achieve accurate PDE solutions under realistic computational constraints. It synthesizes theory (reliability, efficiency, and optimality of estimators and adaptive schemes) with practical engineering practice, GOEE, and nonstandard methods (meshfree, IGA, enrichment) to highlight pathways toward certified numerical simulation and real-time adaptivity. Key contributions include clarifying the reliability and effectivity of estimators, marking strategies like D\'orfler, the goal-oriented paradigm, and extensions to nonlinear and parabolic settings, as well as insights into industrial bottlenecks and real-world applications. The perspective emphasizes bridging rigorous adaptive FEM theory with industrial needs, enabling reliable, efficient, and sometimes real-time simulations across complex multi-physics problems.

Abstract

Error control by means of a posteriori error estimators or indica-tors and adaptive discretizations, such as adaptive mesh refinement, have emerged in the late seventies. Since then, numerous theoretical developments and improvements have been made, as well as the first attempts to introduce them into real-life industrial applications. The present introductory chapter provides an overview of the subject, highlights some of the achievements to date and discusses possible perspectives.

A short perspective on a posteriori error control and adaptive discretizations

TL;DR

The work surveys a posteriori error estimation and adaptive discretization as a framework to achieve accurate PDE solutions under realistic computational constraints. It synthesizes theory (reliability, efficiency, and optimality of estimators and adaptive schemes) with practical engineering practice, GOEE, and nonstandard methods (meshfree, IGA, enrichment) to highlight pathways toward certified numerical simulation and real-time adaptivity. Key contributions include clarifying the reliability and effectivity of estimators, marking strategies like D\'orfler, the goal-oriented paradigm, and extensions to nonlinear and parabolic settings, as well as insights into industrial bottlenecks and real-world applications. The perspective emphasizes bridging rigorous adaptive FEM theory with industrial needs, enabling reliable, efficient, and sometimes real-time simulations across complex multi-physics problems.

Abstract

Error control by means of a posteriori error estimators or indica-tors and adaptive discretizations, such as adaptive mesh refinement, have emerged in the late seventies. Since then, numerous theoretical developments and improvements have been made, as well as the first attempts to introduce them into real-life industrial applications. The present introductory chapter provides an overview of the subject, highlights some of the achievements to date and discusses possible perspectives.
Paper Structure (22 sections, 24 equations, 2 figures)

This paper contains 22 sections, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A few basic notions
  3. Numerical approximation of a mathematical model
  4. Galerkin methods and the discretization error
  5. Error estimators and error control
  6. Error estimators and mesh refinement
  7. The goal-oriented paradigm
  8. Theory of adaptive finite element methods
  9. Optimality of adaptive algorithms
  10. Other recent achievements
  11. Current engineering practice(s)
  12. Accuracy vs. computational cost vs. complexity
  13. Some initiatives
  14. Some bottlenecks
  15. Special topics and industrial applications
  16. ...and 7 more sections

Figures (2)

  • Figure 1: Mathematical modelling and sources of error. This figure showcases the steps involved in mathematical modelling, which is a process used to represent real-world phenomena using mathematical equations. The diagram highlights the different stages of mathematical modelling, including problem formulation, model development, parameter estimation, model validation, and prediction. Additionally, the figure identifies potential sources of error that may impact the accuracy and reliability of the mathematical model. These sources of error can arise from various factors, such as measurement uncertainties, assumptions made during model development, limitations in data availability, simplifications in model assumptions, and uncertainties in parameter estimation. The figure serves as a visual representation of the complexities and challenges associated with mathematical modelling, emphasising the need for careful consideration of potential sources of errors to ensure the robustness and validity of the model's results. It underscores the importance of thorough validation and verification processes to enhance the accuracy and reliability of mathematical models, which are crucial for decision-making, prediction, and understanding complex systems in various fields of science, engineering, and beyond.
  • Figure 2: This figure showcases the process of mathematical modelling along with the identification and estimation of potential sources of error. The diagram illustrates the steps involved in mathematical modelling, including problem formulation, model development, parameter estimation, model validation, and prediction. Furthermore, the figure highlights the importance of error estimation in the modelling process. It identifies potential sources of error, such as measurement uncertainties, assumptions made during model development, limitations in data availability, simplifications in model assumptions, and uncertainties in parameter estimation. The figure emphasises the need to account for and quantify these sources of error in order to assess the reliability and accuracy of the mathematical model. The figure serves as a visual representation of the comprehensive approach to mathematical modelling, which includes not only model development but also thorough error estimation to enhance the robustness and validity of the model's results. It underscores the significance of error estimation in improving the quality of mathematical models and their applicability in various fields of science, engineering, and beyond.

Theorems & Definitions (2)

  • Remark 1
  • Remark 2