WELL-POSEDNESS AND DISCRETIZATION FOR A CLASS OF MODELS FOR MIXED-DIMENSIONAL PROBLEMS WITH HIGH-DIMENSIONAL GAP
WELL-POSEDNESS AND DISCRETIZATION FOR A CLASS OF MODELS FOR MIXED-DIMENSIONAL PROBLEMS WITH HIGH-DIMENSIONAL GAP
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DOI：
10.1137/20m1362541
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发表时间：
2021-01-01
影响因子：
1.9
通讯作者：
Nordbotten, Jan M.
中科院分区：
文献类型：
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作者：
Hodneland, Erlend;Hu, Xiaozhe;Nordbotten, Jan M.
在这项工作中，我们显示了潜在的数学结构的混合维模型所产生的组成图和连续域。这样的模型在应用中变得流行，特别是对人体脉管系统进行建模。我们首先讨论了强形式的模型方程，它描述了连续体和网络中的质量守恒和达西定律以及它们之间的耦合。通过引入适当的尺度，我们提出了一个弱形式，避免退化。弱形式的适定性通过标准的Babusv \ka-Brezzi理论证明。我们还发展了混合列式有限元方法，并证明了它的适定性。质量集中技术被引入到推导两点通量近似（TPFA）型离散，以及由于其在应用中的重要性。基于Babusv \ka-Brezzi理论，给出了有限元格式和TPFA格式的误差估计.我们还讨论了有效的线性离散问题求解器。最后，我们给出了一些数值例子来验证理论结果，并证明我们提出的离散方案的鲁棒性。
In this work, we show the underlying mathematical structure of mixed-dimensional models arising from the composition of graphs and continuous domains. Such models are becoming popular in applications, in particular, to model the human vasculature. We first discuss the model equations in the strong form, which describes the conservation of mass and Darcy's law in the continuum and network as well as the coupling between them. By introducing proper scaling, we propose a weak form that avoids degeneracy. Well-posedness of the weak form is shown through standard Babusv \ka--Brezzi theory. We also develop the mixed formulation finite-element method and prove its well-posedness. A mass-lumping technique is introduced to derive the two-point flux approximation (TPFA) type discretization as well, due to its importance in applications. Based on the Babusv \ka--Brezzi theory, error estimates can be obtained for both the finite-element scheme and the TPFA scheme. We also discuss efficient linear solvers for discrete problems. Finally, we present some numerical examples to verify the theoretical results and demonstrate the robustness of our proposed discretization schemes.