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Adaptivity and blow-up detection for nonlinear evolution problems

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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.

Funding

The work of the first and fourth authors was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) through the First Grant scheme (grant EP/L022745/1) and a doctoral training grant, respectively. The work of the third author was supported in part by the European Social Fund (ESF), European Union (EU), and National Resources of the Greek State within the framework of the Action \Supporting Postdoctoral Researchers" of the Operational Programme \Education and Lifelong Learning (EdLL).

History

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

issn

1064-8275

eissn

1095-7197

Acceptance date

19/09/2016

Copyright date

2016

Available date

13/07/2017

Publisher version

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

Language

en