[Paper Review] Modeling, Structure and Discretization of Mixed-dimensional Partial Differential Equations

Jan M. Nordbotten, Wietse M. Boon|arXiv (Cornell University)|May 19, 2017

Advanced Mathematical Modeling in Engineering5 references3 citations

TL;DR

This paper presents a comprehensive framework for modeling, analyzing, and discretizing mixed-dimensional partial differential equations (PDEs) where sub-domains are one dimension lower than the full domain, such as fractures in porous media or vessels in biological systems. It establishes a mathematical foundation using conservation laws, differential geometry, and variational formulations, leading to a stable, convergent finite element discretization that handles intersecting manifolds and hierarchical dimensionality reduction.

ABSTRACT

Mixed-dimensional partial differential equations arise in several physical applications, wherein parts of the domain have extreme aspect ratios. In this case, it is often appealing to model these features as lower-dimensional manifolds embedded into the full domain. Examples are fractured and composite materials, but also wells (in geological applications), plant roots, or arteries and veins. In this manuscript, we survey the structure of mixed-dimensional PDEs in the context where the sub-manifolds are a single dimension lower than the full domain, including the important aspect of intersecting sub-manifolds, leading to a hierarchy of successively lower-dimensional sub-manifolds. We are particularly interested in partial differential equations arising from conservation laws. Our aim is to provide an introduction to such problems, including the mathematical modeling, differential geometry, and discretization.

Motivation & Objective

• To develop a systematic mathematical framework for mixed-dimensional PDEs arising in physical systems with extreme aspect ratios.
• To address the modeling of sub-domains embedded as lower-dimensional manifolds, particularly in intersecting and hierarchical configurations.
• To analyze the structure of such PDEs using differential geometry and variational formulations.
• To provide a stable and convergent finite element discretization method for mixed-dimensional problems.
• To unify modeling, geometric structure, and numerical discretization in a single coherent treatment for applications in porous media, biology, and engineering.

Proposed method

• Formulates mixed-dimensional PDEs using conservation laws on domains with embedded manifolds of codimension one.
• Applies differential geometry to describe the embedding of lower-dimensional manifolds and their interactions.
• Derives weak formulations via variational principles to ensure mathematical consistency and stability.
• Proposes a finite element method that respects the dimensionality mismatch by using appropriate function spaces on the full and lower-dimensional domains.
• Handles intersecting manifolds through hierarchical decomposition and consistent transmission conditions across interfaces.
• Validates the approach through theoretical convergence analysis and numerical examples on model problems.

Experimental results

Research questions

• RQ1 How can mixed-dimensional PDEs be consistently formulated when sub-domains are one dimension lower than the ambient space?
• RQ2 What geometric and variational structures underlie mixed-dimensional PDEs in the presence of intersecting manifolds?
• RQ3 How can a stable and convergent finite element method be constructed for such problems despite the dimension mismatch?
• RQ4 What are the essential transmission conditions at intersections of lower-dimensional manifolds?
• RQ5 How do conservation laws manifest in mixed-dimensional settings, and how can they be preserved in the discrete formulation?

Key findings

• The paper establishes a consistent variational formulation for mixed-dimensional PDEs derived from conservation laws, ensuring physical consistency.
• It identifies the role of trace operators and jump conditions at manifold interfaces, particularly in intersecting configurations.
• The proposed finite element method achieves optimal convergence rates in appropriate energy norms, as shown through theoretical analysis.
• The framework handles hierarchical and intersecting manifolds by decomposing the problem into nested sub-manifold interactions with consistent coupling.
• Numerical experiments confirm the stability and accuracy of the discretization, even in complex intersection geometries.
• The approach provides a unified treatment of mixed-dimensional problems across applications in porous media, biology, and structural mechanics.