Analysis and approximation of mixed-dimensional PDEs on 3D-1D domains coupled with Lagrange multipliers
Analysis and approximation of mixed-dimensional PDEs on 3D-1D domains coupled with Lagrange multipliers
复制标题
结合拉格朗日乘子的 3D-1D 域上混合维偏微分方程的分析和逼近
DOI：
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发表时间：
2020
影响因子：
2.9
通讯作者：
P. Zunino
中科院分区：
文献类型：
--
作者：
M. Kuchta;F. Laurino;K. Mardal;P. Zunino
定义在不同维数区域上的耦合偏微分方程通常称为混合维偏微分方程。我们解决三维（3D）和一维域上的混合维偏微分方程，从而产生一个3D-1D耦合问题。从解的存在性和数值逼近的角度来看，这样的问题提出了一些挑战。对于跨维耦合条件，我们考虑本质条件和自然条件的组合，基本上是Dirichlet条件和Neumann条件的组合。为了确保一个有意义的制定这样的条件，我们使用的拉格朗日乘子法，适当地适应于混合维的情况下。由此产生的鞍点问题的适定性进行了分析。然后，我们解决的有限元方法的框架中的问题的数值逼近。拉格朗日乘子空间的离散化是主要的挑战。提出了几种选择，分析和比较，目的是确定一个良好的平衡之间的数学性质的离散问题和灵活性的数值方案的实施。基于数值实验的证据支持的结果。
Coupled partial differential equations defined on domains with different dimensionality are usually called mixed dimensional PDEs. We address mixed dimensional PDEs on three-dimensional (3D) and one-dimensional domains, giving rise to a 3D-1D coupled problem. Such problem poses several challenges from the standpoint of existence of solutions and numerical approximation. For the coupling conditions across dimensions, we consider the combination of essential and natural conditions, basically the combination of Dirichlet and Neumann conditions. To ensure a meaningful formulation of such conditions, we use the Lagrange multiplier method, suitably adapted to the mixed dimensional case. The well posedness of the resulting saddle point problem is analyzed. Then, we address the numerical approximation of the problem in the framework of the finite element method. The discretization of the Lagrange multiplier space is the main challenge. Several options are proposed, analyzed and compared, with the purpose to determine a good balance between the mathematical properties of the discrete problem and flexibility of implementation of the numerical scheme. The results are supported by evidence based on numerical experiments.
影响因子：
2.9
作者：
Arioli M
通讯作者：
Arioli M