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# Adaptivity and blow-up detection for nonlinear evolution problems

journal contribution
posted on 13.07.2017, 13:08 by Andrea Cangiani, Emmanuil H. Georgoulis, Irene Kyza, Stephen Metcalfe
This work is concerned with the development of a space-time adaptive numerical method, based on a rigorous a posteriori error bound, for a semilinear convection-diffusion problem which may exhibit blow-up in finite time. More specifically, a posteriori error bounds are derived in the $L^{\infty}(L^2)+L^2(H^1)$-type norm for a first order in time implicit-explicit interior penalty discontinuous Galerkin in space discretization of the problem, although the theory presented is directly applicable to the case of conforming finite element approximations in space. The choice of the discretization in time is made based on a careful analysis of adaptive time-stepping methods for ODEs that exhibit finite time blow-up. The new adaptive algorithm is shown to accurately estimate the blow-up time of a number of problems, including one which exhibits regional blow-up.

## Citation

SIAM Journal on Scientific Computing, 2016, 38 (6), pp. A3833-A3856

## Author affiliation

/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of Mathematics

## Version

VoR (Version of Record)

## Published in

SIAM Journal on Scientific Computing

## Publisher

SIAM PUBLICATIONS

1064-8275

1095-7197

19/09/2016

2016

13/07/2017

## Publisher version

http://epubs.siam.org/doi/abs/10.1137/16M106073X

en