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.
- finite time blow-up
- conditional a posteriori error estimates
- IMEX method
- discontinuous Galerkin methods
Cangiani, A., Georgoulis, E. H., Kyza, I., & Metcalfe, S. (2016). Adaptivity and blow-up detection for nonlinear evolution problems. SIAM Journal on Scientific Computing, 38(6), A3833-A3856. https://doi.org/10.1137/16M106073X