Reduced Lagrange multiplier approach for non-matching coupling of mixed-dimensional domains

TL;DR

The paper addresses the challenge of coupling PDEs defined on domains of different dimensions with non matching interfaces by introducing a reduced Lagrange multiplier framework. By employing restriction and extension operators, and a Fourier based extension for cylindrical inclusions, it analyzes stability, well posedness, and dimensionality reduction error of the reduced saddle point problem, and provides a priori error estimates for its numerical discretization. It reveals a precise interplay between the mesh size $h$, the number of multiplier modes $N$ (or equivalently the cross sectional modes $n$), and the inclusion size $ε$, showing how to balance modeling and discretization errors while achieving accurate, efficient coupling in both 2D and 3D settings. The work presents extensive numerical experiments validating the theory and guiding practical choices for the mode count and mesh resolution in applications such as fractured media, fiber reinforced materials, and microcirculation. Overall, the reduced Lagrange multiplier approach provides a rigorous and adaptable framework for non matching mixed dimensional coupling with controllable error sources and scalable numerical schemes.

Abstract

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.

Figures (11)

• Figure 1: (left panel) Geometry of randomly placed cylindrical fibers in a three-dimensional continuum. The cylinders have radius $\epsilon=0.2$ and height $\eta=0.5$ are placed randomly in a non-overlapping way and at a finite distance from the boundary of the domain. (right panel) Example of the two dimensional section of a single one fiber, corresponding to the domain $V$ of radius $\epsilon$, embedded in a macroscopic domain $\Omega$.
• Figure 2: Example of dimensionality reduction from a general inclusion $V$ with boundary $\Gamma\equiv\gamma$ to a representative subset $\gamma$.
• Figure 3: Example for the numerical solution with global refinements for problem \ref{['eq:manufactured-solution-D1']} (left) and with local refinements (right), using a single Fourier mode and $r=0.2$.
• Figure 4: Error of the numerical solution with global refinements (left) and with local refinements (right), using a single Fourier mode for problem \ref{['eq:manufactured-solution-D1']}, when $r=0.2$.
• Figure 5: Example for a numerical solution where the dimensionality reduction error is significant, using a single Fourier mode (left) and two fourier modes (right), for problem \ref{['eq:manufactured-solution-D2']}, when $r=0.2$.

Theorems & Definitions (10)

• Theorem 1: Infsup on $A$
• Theorem 2: Infsup on $B$
• Remark 1
• Lemma 1: Infsup on $RB$
• Remark 2: Examples
• Lemma 2: regularity of reference weighted operators
• Theorem 3: regularity of weighted operators
• Theorem 4: Modal extension operator
• Lemma 3
• Lemma 4