The objective of this paper is to develop efficient numerical algorithms for the linear advection-diffusion equation in fractured porous media. A reduced fracture model is considered where the fractures are treated as interfaces between subdomains and the interactions between the fractures and the surrounding porous medium are taken into account. The model is discretized by a backward Euler upwind-mixed hybrid finite element method in which the flux variable represents both the advective and diffusive fluxes. The existence, uniqueness, as well as optimal error estimates in both space and time for the fully discrete coupled problem are established. Moreover, to facilitate different time steps in the fracture-interface and the subdomains, global-in-time, nonoverlapping domain decomposition is utilized to derive two implicit iterative solvers for the discrete problem. The first method is based on the time-dependent Steklov–Poincaré operator, while the second one employs the optimized Schwarz waveform relaxation (OSWR) approach with Ventcel-Robin transmission conditions. A discrete space-time interface system is formulated for each method and is solved iteratively with possibly variable time step sizes. The convergence of the OSWR-based method with conforming time grids is also proved. Finally, numerical results in two dimensions are presented to verify the optimal order of convergence of the monolithic solver and to illustrate the performance of the two decoupled schemes with local time-stepping on problems of high Péclet numbers.

中文翻译：

---

用于裂缝多孔介质输运问题的整体和局部时间步进解耦算法

本文的目的是为裂缝多孔介质中的线性平流扩散方程开发有效的数值算法。考虑简化裂缝模型，其中裂缝被视为子域之间的界面，并考虑裂缝与周围多孔介质之间的相互作用。该模型采用后向欧拉迎风混合混合有限元方法进行离散化，其中通量变量代表平流通量和扩散通量。建立了全离散耦合问题的存在性、唯一性以及空间和时间上的最优误差估计。此外，为了促进断裂界面和子域中的不同时间步长，利用全局时间、非重叠域分解来导出离散问题的两个隐式迭代求解器。第一种方法基于时间相关的 Steklov-Poincaré 算子，而第二种方法则采用具有 Ventcel-Robin 传输条件的优化 Schwarz 波形弛豫 (OSWR) 方法。为每种方法制定离散时空界面系统，并以可能可变的时间步长迭代求解。还证明了基于 OSWR 的方法在符合时间网格的情况下的收敛性。最后，给出了二维数值结果，以验证整体求解器的最佳收敛阶数，并说明两种具有局部时间步长的解耦方案在高佩克莱特数问题上的性能。