## Seminar in Numerical Analysis: Martin Vohralik (Inria Paris)

**A posteriori error estimates robust with respect to nonlinearities and final time**

A posteriori estimates enable to certify the error committed in a numerical simulation. In particular, the equilibrated flux reconstruction technique yields a guaranteed error upper bound, where the flux, obtained by a local postprocessing, is of independent interest since it is always locally conservative. In this talk, we tailor this methodology to model nonlinear and time-dependent problems to obtain estimates that are robust, i.e., of quality independent of the strength of the nonlinearities and the final time. These estimates include, and build on, common iterative linearization schemes such as Zarantonello, Picard, Newton, or M- and L-ones. We first consider steady problems and conceive two settings: we either augment the energy difference by the discretization error of the current linearization step, or we design iteration-dependent norms that feature weights given by the current iterate. We then turn to unsteady problems. Here we first consider the linear heat equation and finally move to the Richards one, that is doubly nonlinear and exhibits both parabolic–hyperbolic and parabolic–elliptic degeneracies. Robustness with respect to the final time and local efficiency in both time and space are addressed here. Numerical experiments illustrate the theoretical findings all along the presentation. Details can be found in [1-4].

A. Ern, I. Smears, M. Vohralík, Guaranteed, locally space-time efficient, and polynomial-degree robust a posteriori error estimates for high-order discretizations of parabolic problems, *SIAM J. Numer. Anal.***55** (2017), 2811–2834.

A. Harnist, K. Mitra, A. Rappaport, M. Vohralík, Robust energy a posteriori estimates for nonlinear elliptic problems, HAL Preprint 04033438, 2023.

K. Mitra, M. Vohralík, A posteriori error estimates for the Richards equation, *Math. Comp.* (2024), accepted for publication.

K. Mitra, M. Vohralík, Guaranteed, locally efficient, and robust a posteriori estimates for nonlinear elliptic problems in iteration-dependent norms. An orthogonal decomposition result based on iterative linearization, HAL Preprint 04156711, 2023.

For further information about the seminar, please visit this webpage.

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