非线性演化问题的适应性与爆炸检测
Adaptivity and blow-up detection for nonlinear evolution problems
AI总结：
本文提出一种时空自适应数值方法，用于解决可能在有限时间内出现爆炸的半线性对流扩散问题，通过严格后验误差界实现高精度爆炸时间估计。
摘要：
本文提出了一种时空自适应数值方法，基于严格的后验误差界，用于解决可能在有限时间内出现爆炸的半线性对流扩散问题。更具体地说，推导了$L^{\infty}(L^2)+L^2(H^1)$型范数下的后验误差界，针对一个问题的一阶时间隐式-显式（IMEX）内部惩罚不连续伽辽金（dG）空间离散化方法。尽管所提出的理论直接适用于符合有限元近似的情况，但时间离散化的选择是基于对ODEs中有限时间爆炸适应性时间步方法的仔细分析。新的自适应算法被证明能够准确估计多个问题的爆炸时间，包括一个表现出区域爆炸的问题。
英文摘要：
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 (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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
