In this article, finite element a posteriori error estimates for the linear parabolic integro-differential equation using the two-step backward time descretization formula are explored. For space discretization, we use piecewise linear finite element spaces. The Ritz–Volterra reconstruction operator is used as a raw ingredient to obtain the optimal convergence in space. Further, a novel quadratic space–time reconstruction operator, namely BDF2 operator, is introduced to achieve second-order accuracy in time. Numerical results demonstrate the theoretical findings.

中文翻译：

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抛物线积分微分方程的后验误差估计是完全离散的，采用了两步向后微分公式

本文研究了线性抛物线积分微分方程的后验误差估计，采用了两步倒数时间降噪公式。对于空间离散化，我们使用分段线性有限元空间。Ritz–Volterra重建算子被用作原始成分，以获得最佳的空间收敛性。此外，引入了一种新颖的二次时空重构算子，即BDF2算子，以实现时间的二阶精度。数值结果证明了理论发现。