We study the construction and convergence of decoupling multistep schemes of higher order using the backward differentiation formulae for an elliptic-parabolic problem, which includes multiple-network poroelasticity as a special case. These schemes were first introduced in [Altmann, Maier, Unger, BIT Numer. Math., 64:20, 2024], where a convergence proof for the second-order case is presented. Here, we present a slightly modified version of these schemes using a different construction of related time delay systems. We present a novel convergence proof relying on concepts from G-stability applicable for any order and providing a sharper characterization of the required weak coupling condition. The key tool for the convergence analysis is the construction of a weighted norm enabling a telescoping argument for the sum of the errors.

翻译：本文研究针对椭圆-抛物问题（以多网络多孔弹性问题为特例）采用向后差分公式的高阶解耦多步格式的构造与收敛性。此类格式最初由[Altmann, Maier, Unger, BIT Numer. Math., 64:20, 2024]提出，其中给出了二阶情形的收敛性证明。本文通过构建不同的关联时滞系统，提出了这些格式的改进版本。我们提出了一种基于G-稳定性概念的新型收敛性证明方法，该方法适用于任意阶格式，并对所需的弱耦合条件给出了更精确的刻画。收敛性分析的关键工具是构建加权范数，从而实现对误差和的可伸缩论证。