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.

中文翻译：

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半线性热方程中爆破的逐点后验误差界限

这项工作涉及时空自适应数值方法的开发，该方法基于严格的后验误差界限，用于具有一般局部 Lipschitz 反应项的半线性热方程，其解可能在有限时间内爆炸。更具体地说，条件后验误差界是在 $L^{\infty}L^{\infty}$ 范数中导出的，用于时间上的一阶，隐式 - 显式（IMEX），符合有限元方法在空间离散化问题。适用于爆炸和非爆炸案例的数值实验突出了我们方法的普遍性并补充了理论结果。