Conditional a posteriori error bounds for high order DG time stepping approximations of semilinear heat models with blow-up

This work is concerned with the development of an adaptive numerical method for semilinear heat flow models featuring a general (possibly) nonlinear reaction term that may cause the solution to blow up in finite time. The fully discrete scheme consists of a high order discontinuous Galerkin (dG) time stepping method and a conforming finite element discretisation (cG) in space. The proposed adaptive procedure is based on rigorously devised conditional a posteriori error bounds in the $L^{\mathrm{\infty }}(L^{\mathrm{\infty }})$ norm. Numerical experiments complement the theoretical results.