In this paper, we are interested in the mathematical properties of methods based on a fictitious domain approach combined with reduced-order interface coupling conditions, which have been recently introduced to simulate 3D-1D fluid-structure or structure-structure coupled problems. To give insights on the approximation properties of these methods, we investigate them in a simplified setting by considering the Poisson problem in a two-dimensional domain with non-homogeneous Dirichlet boundary conditions on small inclusions. The approximated reduced problem is obtained using a fictitious domain approach combined with a projection on a Fourier finite-dimensional space of the Lagrange multiplier associated to the Dirichlet boundary constraints, obtaining in this way a Poisson problem with defective interface conditions. After analyzing the existence of a solution of the reduced problem, we prove its convergence towards the original full problem, when the size of the holes tends towards zero, with a rate which depends on the number of modes of the finite-dimensional space. In particular, our estimates highlight the fact that to obtain a good convergence on the Lagrange multiplier, one needs to consider more modes than the first Fourier mode (constant mode). This is a key issue when one wants to deal with real coupled problems, such as fluid-structure problems for instance. Next, the numerical discretization of the reduced problem using the finite element method is analyzed in the case where the computational mesh does not fit the small inclusion interface. As is standard for these types of problem, the convergence of the solution is not optimal due to the lack of regularity of the solution. Moreover, convergence exhibits a well-known locking effect when the mesh size and the inclusion size are of the same order of magnitude. This locking effect is more apparent when increasing the number of modes and affects the Lagrange multiplier convergence rate more heavily. To resolve these issues, we propose and analyze a stabilized method and an enriched method for which additional basis functions are added without changing the approximation space of the Lagrange multiplier. Finally, the properties of numerical strategies are illustrated by numerical experiments.

中文翻译：

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混合维问题的降阶界面条件和增广有限元的数学和数值分析


在本文中，我们对基于虚构域方法结合降阶界面耦合条件的方法的数学性质感兴趣，这些方法最近被引入用于模拟 3D-1D 流-固或结构-结构耦合问题。为了深入了解这些方法的近似特性，我们通过在二维域中考虑小夹杂物上具有非齐次狄利克雷边界条件的泊松问题，在简化的环境中研究它们。近似约简问题是使用虚构域方法结合与狄利克雷边界约束相关的拉格朗日乘子在傅里叶有限维空间上的投影获得的，以这种方式获得界面条件有缺陷的泊松问题。在分析了约化问题解的存在性后，我们证明了它向原始完整问题的收敛，当空穴的大小趋于零时，其速率取决于有限维空间的模式数。特别是，我们的估计强调了这样一个事实，即要在拉格朗日乘子上获得良好的收敛性，需要考虑比第一个傅里叶模式（常数模式）更多的模式。当人们想要处理真正的耦合问题时，例如流体结构问题，这是一个关键问题。接下来，在计算网格不适合小夹杂物界面的情况下，使用有限元方法分析约化问题的数值离散化。与这类问题的标准一样，由于解缺乏规律性，解的收敛性不是最优的。 此外，当网格尺寸和夹杂物尺寸相同数量级时，收敛表现出众所周知的锁定效应。当增加模式数量时，这种锁定效果更为明显，并且对拉格朗日乘子收敛速率的影响更大。为了解决这些问题，我们提出并分析了一种稳定方法和一种富集方法，其中添加了额外的基函数，而不改变拉格朗日乘子的近似空间。最后，通过数值实验说明了数值策略的性质。