We prove first-order convergence of the semi-explicit Euler scheme combined with a finite element discretization in space for elliptic-parabolic problems which are weakly coupled. This setting includes poroelasticity, thermoelasticity, as well as multiple-network models used in medical applications. The semi-explicit approach decouples the system such that each time step requires the solution of two small and well-structured linear systems rather than the solution of one large system. The decoupling improves the computational efficiency without decreasing the convergence rates. The presented convergence proof is based on an interpretation of the scheme as an implicit method applied to a constrained partial differential equation with delay term. Here, the delay time equals the used step size. This connection also allows a deeper understanding of the weak coupling condition, which we accomplish to quantify explicitly.

中文翻译：

---

弱耦合椭圆抛物线问题的半显式离散化方案

对于弱耦合的椭圆抛物线问题，我们证明了半显式欧拉方案与空间有限元离散化相结合的一阶收敛性。此设置包括多孔弹性、热弹性以及医疗应用中使用的多网络模型。半显式方法将系统解耦，使得每个时间步长都需要两个小型且结构良好的线性系统的解决方案，而不是一个大型系统的解决方案。解耦在不降低收敛速度的情况下提高了计算效率。所提出的收敛证明基于将该方案解释为应用于具有延迟项的约束偏微分方程的隐式方法。这里，延迟时间等于使用的步长。