A class of linear parabolic equations are considered. We derive a common framework for the a posteriori error analysis of certain second-order time discretisations combined with finite element discretisations in space. In particular we study the Crank-Nicolson method, the extrapolated Euler method, the backward differentiation formula of order 2 (BDF-2), the Lobatto IIIC method and a two-stage SDIRK method. We use the idea of elliptic reconstructions and certain bounds for the Green's function of the parabolic operator.

翻译：考虑一类线性抛物型方程。针对特定二阶时间离散化与空间有限元离散化相结合的情形，我们推导出一个适用于后验误差分析的通用框架。具体研究了Crank-Nicolson方法、外推欧拉法、二阶向后差分公式（BDF-2）、Lobatto IIIC方法以及两阶段SDIRK方法。我们采用椭圆重构思想以及抛物型算子格林函数的特定有界性条件。