DMI, Spiegelgasse 5, 4051 Basel Seminarraum 05.001

## Seminar in Numerical Analysis: Andreas Veeser (University of Milan)

**Strictly equivalent a posteriori error estimators**

In the context of finite element methods for boundary value problems, the goal of an a posteriori error analysis is to derive a so-called error estimator. Such an estimator should be a quantity that is computable in terms of the problem data and the finite element solution and equivalent to its error. Almost all error estimators available in literature, however, do not meet these requirements. Indeed, equivalence is typically verified only up to so-called (data) oscillations and their presence has been viewed as the price of computability.

The talk will consist of two parts. The first part will argue that the infinite-dimensional nature of the problem data is an obstruction to computability, while the typical form of oscillations obstructs equivalence. In other words: the desirable but missing unspoiled equivalence is not only a technical problem but does not hold for the involved oscillations.

The second part will then present the new approach to a posteriori error estimation proposed by [1]. Its resulting estimators are equivalent to the error and consist of two parts, where the first one is of finite-dimensional nature and thus computable, while the second one is a new form of data oscillation, which is always smaller that the old one and whose computability hinges on the knowledge of the problem data. This splitting of the error estimator is also convenient in guiding adaptive algorithms; cf. [2].

[1] C. Kreuzer, A. Veeser, Oscillation in a posteriori error estimation, Numer. Math. 148 (2021), 43-78

[2] A. Bonito, C. Canuto, R. H. Nochetto, A. Veeser, Adaptive finite element methods, Acta Numerica 33 (2024), 163-485.

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