Reduced Lagrange multiplier approach for non-matching coupling of mixed-dimensional domains
arxiv(2023)
摘要
Many physical problems involving heterogeneous spatial scales, such as the
flow through fractured porous media, the study of fiber-reinforced materials,
or the modeling of the small circulation in living tissues – just to mention a
few examples – can be described as coupled partial differential equations
defined in domains of heterogeneous dimensions that are embedded into each
other. This formulation is a consequence of geometric model reduction
techniques that transform the original problems defined in complex
three-dimensional domains into more tractable ones. The definition and the
approximation of coupling operators suitable for this class of problems is
still a challenge. We develop a general mathematical framework for the analysis
and the approximation of partial differential equations coupled by non-matching
constraints across different dimensions, focusing on their enforcement using
Lagrange multipliers. In this context, we address in abstract and general terms
the well-posedness, stability, and robustness of the problem with respect to
the smallest characteristic length of the embedded domain. We also address the
numerical approximation of the problem and we discuss the inf-sup stability of
the proposed numerical scheme for some representative configuration of the
embedded domain. The main message of this work is twofold: from the standpoint
of the theory of mixed-dimensional problems, we provide general and abstract
mathematical tools to formulate coupled problems across dimensions. From the
practical standpoint of the numerical approximation, we show the interplay
between the mesh characteristic size, the dimension of the Lagrange multiplier
space, and the size of the inclusion in representative configurations
interesting for applications. The latter analysis is complemented with
illustrative numerical examples.
更多查看译文
关键词
coupling,reduced lagrange,multiplier approach,non-matching,mixed-dimensional
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络
Chat Paper
正在生成论文摘要