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DTSTAMP:20231127T102321
DTSTART;TZID=Europe/Zurich:20231208T110000
DTEND;TZID=Europe/Zurich:20231208T120000
SUMMARY:Seminar in Numerical Analysis: Martin Vohralik (Inria Paris)
LOCATION:
DESCRIPTION:A posteriori estimates enable to certify the error committed in
a numerical simulation. In particular, the equilibrated flux r
econstruction technique yields a guaranteed error upper bound,
where the flux, obtained by a local postprocessing, is of indep
endent interest since it is always locally conservative. In thi
s talk, we tailor this methodology to model nonlinear and time-
dependent problems to obtain estimates that are robust, i.e., o
f 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, N
ewton, 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 giv
en by the current iterate. We then turn to unsteady problems. H
ere we first consider the linear heat equation and finally move
to the Richards one, that is doubly nonlinear and exhibits bot
h 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 experimen
ts illustrate the theoretical findings all along the presentati
on. Details can be found in [1-4].\r\nA. Ern, I. Smears, M. Voh
ralík, Guaranteed, locally space-time efficient, and polynomia
l-degree robust a posteriori error estimates for high-order dis
cretizations of parabolic problems, SIAM J. Numer. Anal. 55 (20
17), 2811–2834.\r\nA. Harnist, K. Mitra, A. Rappaport, M. Voh
ralík, Robust energy a posteriori estimates for nonlinear elli
ptic problems, HAL Preprint 04033438, 2023.\r\nK. Mitra, M. Vo
hralík, A posteriori error estimates for the Richards equation
, Math. Comp. (2024), accepted for publication.\r\nK. Mitra, M.
Vohralík, Guaranteed, locally efficient, and robust a posteri
ori estimates for nonlinear elliptic problems in iteration-depe
ndent norms. An orthogonal decomposition result based on iterat
ive linearization, HAL Preprint 04156711, 2023.\r\n \r\nFor f
urther information about the seminar, please visit this webpage
[https://dmi.unibas.ch/de/forschung/mathematik/seminar-in-nume
rical-analysis/].
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