Adaptivity and Blow-Up Detection for Nonlinear Evolution Problems
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 $L^{\infty}(L^2)+L^2(H^1)$-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.
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Information & Authors
Information
Published In
SIAM Journal on Scientific Computing
Pages: A3833 - A3856
DOI: 10.1137/16M106073X
ISSN (online): 1095-7197
Copyright
© 2016, Society for Industrial and Applied Mathematics.
History
Submitted: 8 February 2016
Accepted: 19 September 2016
Published online: 20 December 2016
Keywords
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Authors
Funding Information
European Social Fund https://doi.org/10.13039/501100004895 : EdLL
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/L022745/1
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