SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 558-582, January 2021.
Coupled partial differential equations (PDEs) defined on domains with different dimensionality are usually called mixed-dimensional PDEs. We address mixed-dimensional PDEs on three-dimensional (3D) and one-dimensional (1D) domains, which gives rise to a 3D-1D coupled problem. Such a 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, which are 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 of determining 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.

中文翻译：

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与Lagrange乘子耦合的3D-1D域上混合尺寸PDE的分析和逼近

SIAM数值分析学报，第59卷，第1期，第558-582页，2021年1月。
在具有不同维数的域上定义的耦合偏微分方程（PDE）通常称为混合维PDE。我们在三维（3D）和一维（1D）域上处理混合维PDE，这引起了3D-1D耦合问题。从解的存在和数值逼近的角度来看，这样的问题提出了若干挑战。对于跨维度的耦合条件，我们考虑基本条件和自然条件的组合，这些条件基本上是Dirichlet和Neumann条件的组合。为了确保有意义地表达这些条件，我们使用了适合于混合尺寸情况的拉格朗日乘数法。分析了由此产生的鞍点问题的适定性。然后，我们在有限元方法的框架内解决问题的数值近似问题。拉格朗日乘数空间的离散化是主要挑战。提出，分析和比较了几种选择，目的是确定离散问题的数学特性与数值方案实施的灵活性之间的良好平衡。结果得到了基于数值实验的证据的支持。