Pointwise a posteriori error bounds for blow-up in the semilinear heat equation
Provides rigorous error control for adaptive simulations of blow-up phenomena in reaction-diffusion equations, benefiting numerical analysts and computational scientists.
Developed a space-time adaptive numerical method with rigorous a posteriori error bounds for the semilinear heat equation, handling both blow-up and non-blow-up cases. Numerical experiments demonstrate the method's effectiveness.
This work is concerned with the development of a space-time adaptive numerical method, based on a rigorous a posteriori error bound, for the semilinear heat equation with a general local Lipschitz reaction term whose solution may blow-up in finite time. More specifically, conditional a posteriori error bounds are derived in the $L^{\infty}L^{\infty}$ norm for a first order in time, implicit-explicit (IMEX), conforming finite element method in space discretization of the problem. Numerical experiments applied to both blow-up and non blow-up cases highlight the generality of our approach and complement the theoretical results.