论文标题
一类模型的建模，适合和离散化的模型，用于具有高维间隙的混合维度问题
Modeling, well-posedness and discretization for a class of models for mixed-dimensional problems with high dimensional gap
论文作者
论文摘要
在这项工作中，我们说明了由图和连续域的组成产生的混合模型的基本数学结构。这种模型在应用中尤其是为了建模人类脉管系统而变得流行。我们首先以强烈的形式讨论模型方程，该方程描述了质量和达西定律在连续和网络中的保存以及它们之间的耦合。通过引入适当的缩放，我们提出了一种避免退化的弱形式。通过标准的Babuška-Brezzi理论显示了弱形式的良好性。我们还开发了混合配方的有限元方法，并证明了其适应性。引入了一种质量倾斜技术，以得出两点通量近似类型的离散化，这是由于其在应用中的重要性。基于Babuška-Brezzi理论，可以为有限元方案和TPFA方案获得错误估计。我们还讨论有效的线性求解器，以解决离散问题。最后，我们提出了一些数值示例，以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 illustrate 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 Babuška-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 type discretization as well, due to its importance in applications. Based on the Babuška-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.
