Adaptivity and blow-up detection for nonlinear evolution problems

Andrea Cangiani, Emmanuil H. Georgoulis, Irene Kyza, Stephen Metcalfe

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)
197 Downloads (Pure)

Abstract

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∞(L2) + L2(H1)-type norm for a first order in time implicit-explicit (IMEX) interior penalty discontinuous Galerkin (dG) 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.
Original languageEnglish
Pages (from-to)A3833-A3856
Number of pages24
JournalSIAM Journal on Scientific Computing
Volume38
Issue number6
Early online date20 Dec 2016
DOIs
Publication statusPublished - 2016

Keywords

  • finite time blow-up
  • conditional a posteriori error estimates
  • IMEX method
  • discontinuous Galerkin methods

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