Summer school on "Adaptive methods and a posteriori error estimation"

24 June 2024 - 27 June 2024
(4 x 8h)
campus Hasselt


Numerous mathematical models of high societal and technological relevance are (systems of) partial differential equations (PDEs). In the wide majority of cases of interest, finding an explicit solution is not possible, so one needs to use numerical methods. In this case, one has to guarantee that, on the one hand, the numerical approximation is accurate enough, and, on the other hand, the numerical method is efficient, limiting the computations to the minimum necessary.

Some important questions in this regard are:

  • How large is the overall numerical simulation error (the difference between the exact solution and its approximation)?
  • Where is the error localized (in space and time)?
  • Which part/component is responsible for the error: numerical method, mathematical model, regularization, linearization, or the approximate solution of linear algebraic systems?
  • How can the numerical method be adapted to reduce the error (where to consider a finer mesh, and where a higher order scheme)?
  • How can the mathematical model, regularization, linearization, and algebraic resolution be adapted? In particular, how to design iterative schemes to ensure that the iteration error is reduced below a given tolerance, and with a minimal number of iterations?
In the a priori error analysis, a bound on the error is provided from the details of the numerical method without using the approximate solution. This is used to justify the method in the sense that an increase in the computational effort will lead to higher precision. However, these error estimates cannot be used to answer the first question above, as they are not sharp, have a global character and involve various unknown constants. Moreover, they do not shed light on the other four questions.
This is why, in the past decades much attention was paid to a posteriori error analysis. In this case, the approximation itself is used to derive bounds that are fully computable, very precise, estimate the errors locally, and allow to distinguish between different error components. In this way, efficient and adaptive solvers can be developed.
In this summer school, young researches will learn the theoretical background of the a posteriori analysis, and apply these tools to design adaptive solvers for different types of models. This will provide the participants the necessary knowledge to apply these novel techniques to their own research projects.
It will be open to participants having a knowledge in partial differential equations, numerical methods and scientific computing, as they are taught to engineers and scientists. All sessions will be hands-on, meaning that sufficient time for exercises and their discussion is planned.
The course will be centered around the following elements:
  • Error Control
  • Error localization
  • Error components separation
  • Adaptivity

Tentative program
  • Day 1:
    • Basic notions of the a posteriori error analysis (Martin Vohralik/INRIA Paris)
    • Starting with the finite element approximation of linear elliptic problems, the difference between a priori and a posteriori analysis is explained, the Prager-Synge inequality is presented. Also, it is shown how to use the residuals to develop fully computable error indicators.
    • Poster session and reception (complimentary to all participants)
  • Day 2:
    • Extend the knowledge acquired in the first day (Martin Vohralik/INRIA Paris)
    • More general linear problems, mixed finite elements, linear solvers, and adaptivity for linear problems will be considered.
  • Day 3:
    • Nonlinear problems (Koondi Mitra/Eindhoven)
    • The focus will be on iterative solvers, and in particular the adaptive choice of the scheme (a switch between Newton, which converges rapidly, but is very restrictive w.r.t. the initial guess, and more robust, fixed-point approaches), and of the parameters.
    • Group dinner (complimentary to all participants)
  • Day 4:
    • Complex Systems (Elyes Ahmed/SINTEF Digital, Oslo)
    • The presentation will be a guided tour on error control, adaptive time stepping, stopping criteria, applied to domain decomposition, reservoir models, and mixed-dimensional systems
    • The focus will be on non-conforming schemes, simple reconstructions, and their applications for efficient solvers of porous media flows.
The theory will be accompanied by hands-on sessions where students learn to implement this into a (prepared) FreeFem++ framework for the Laplace equation, and straightforward extensions of it.

Learning outcomes

After attending this summer school, participants should be able to:

  • have learned the theoretical background of a posteriori analysis
  • apply these tools to design adaptive solvers for different types of models
  • apply these novel techniques to their own research projects.


An important part of preparing for any further professional step is becoming (more) aware of the competences you have developed and/or want to develop. In the current workshop, the following competences from the UHasselt competency overview are actively dealt with:

  • academic research competences:
    • research methods
    • subject knowledge
  • intellectual competences:
    • analytical thinking
    • problem solving
  • interpersonal competences:
    • presentation skills
    • oral communication


For whom?

  • PhD students and postdocs in (applied) mathematicians, engineering and sciences.

When and where?

  • June 24-27, 2024
  • campus Hasselt, Old prison, room:
    • 24/06/24 - FR-1.05
    • 25/06/24 - FR-2.03
    • 26/06/24 - FR-1.17
    • 27/06/24 - FR-1.05


  • Registration will be possible here as from March 21 until May 25, 2024.
  • As places are limited, registering does not automatically imply that you will be able to participate. You will be notified by email after the registration deadline has passed.
  • Please cancel your registration at least one week in advance in case you cannot make it (cf. cancellation & no-show policy).

Acknowledged as?

  • DS S&T: category 'advanced discipline-specific course'


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