Abstract We discuss two linearization and domain decomposition methods for mathematical models for two-phase flow in a porous medium. The medium consists of two adjacent regions with possibly different parameterizations. The model accounts for non-equilibrium effects like dynamic capillarity and hysteresis. The θ -scheme is adopted for the temporal discretization of the equations yielding nonlinear time-discrete equations. For these, we propose and analyze two iterative schemes, which combine a stabilized linearization iteration of fixed-point type, the L-scheme, and a non-overlapping domain decomposition method into one iteration. First, we prove the existence of unique solutions to the problems defining the linear iterations. Then, we give the rigorous convergence proof for both iterative schemes towards the solution of the time-discrete equations. The developed schemes are independent of the spatial discretization or the mesh and avoid the use of derivatives as in Newton based iterations. Their convergence holds independently of the initial guess, and under mild constraints on the time step. The numerical examples confirm the theoretical results and demonstrate the robustness of the schemes. In particular, the second scheme is well suited for models incorporating hysteresis. The schemes can be easily implemented for realistic applications.

中文翻译：

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涉及动态毛细作用和滞后作用的两相多孔介质流动模型的线性域分解方法

摘要 我们讨论了多孔介质中两相流数学模型的两种线性化和域分解方法。介质由可能具有不同参数化的两个相邻区域组成。该模型考虑了非平衡效应，如动态毛细作用和滞后。θ-方案被用于方程的时间离散化，产生非线性时间离散方程。对于这些，我们提出并分析了两种迭代方案，它们将定点类型的稳定线性化迭代、L 方案和非重叠域分解方法组合为一次迭代。首先，我们证明了定义线性迭代的问题的唯一解的存在。然后，我们为求解时间离散方程的两种迭代方案给出了严格的收敛证明。所开发的方案独立于空间离散化或网格，并避免使用基于牛顿迭代的导数。它们的收敛性独立于初始猜测，并且在时间步长的轻微限制下成立。数值例子证实了理论结果并证明了方案的鲁棒性。特别是，第二种方案非常适合包含滞后的模型。这些方案可以很容易地用于实际应用。并且在时间步长的轻微限制下。数值例子证实了理论结果并证明了方案的鲁棒性。特别是，第二种方案非常适合包含滞后的模型。这些方案可以很容易地用于实际应用。并且在时间步长的轻微限制下。数值例子证实了理论结果并证明了方案的鲁棒性。特别是，第二种方案非常适合包含滞后的模型。这些方案可以很容易地用于实际应用。