Adaptivity and Blow-Up Detection for Nonlinear Evolution Problems

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

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.

• Journal Article Type: Article
• Acceptance Date: Sep 19, 2016
• Online Publication Date: Dec 20, 2016
• Publication Date: 2016
• Deposit Date: Aug 9, 2019
• Journal: SIAM Journal on Scientific Computing
• Print ISSN: 1064-8275
• Electronic ISSN: 1095-7197
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 38
• Issue: 6
• Pages: A3833-A3856
• DOI: https://doi.org/10.1137/16m106073x
• Public URL: https://nottingham-repository.worktribe.com/output/2411738
• Publisher URL: https://epubs.siam.org/doi/10.1137/16M106073X