Derivation and analysis of coupled PDEs on manifolds with high dimensionality gap arising from topological model reduction

Abstract

Multiscale methods based on coupled partial differential equations defined on bulk and embedded manifolds are still poorly explored from the theoretical standpoint, although they are successfully used in applications, such as microcirculation and flow in perforated subsurface reservoirs. This work aims at shedding light on some theoretical aspects of a multiscale method consisting of coupled partial differential equations defined on one-dimensional domains embedded into three-dimensional ones. Mathematical issues arise because the dimensionality gap between the bulk and the inclusions is larger than one, that is the high dimensionality gap case. First, we show that such model derives from a system of fully three-dimensional equations, by the application of a topological model reduction approach. Secondly, we rigorously analyze the problem, showing that the averaging operators applied for the model reduction introduce a regularization effect that resolves the issues due to the singularity of solutions and to the ill-posedness of restriction operators. Then, we exploit the structure of the model reduction technique to analyze the modeling error. This study confirms that for infinitesimally small inclusions, the modeling error vanishes. Finally, we discretize the problem by means of the finite element method and we analyze the approximation and the model error by means of numerical experiments.

Whither Magnetic Hyperthermia? A Tentative Roadmap

“…Due to its inherent complexity, this is a longstanding problem that needs to be addressed in the short-term to move towards wider clinical adoption of magnetic hyperthermia. A unified magnetic hyperthermia theory should be made possible in the near future, taking advantage of 3D-1D coupling strategies based on topological model reduction [189].…”

A mixed framework for topological model reduction of coupled PDEs

“…The singularity issue was remedied in a series of papers by Köppl and coauthors [12,15,4], in which the authors considered an alternative coupling of the model. This idea was further developed by Laurino and Zunino [16], where the coupled 1D-3D flow model was rigorously rederived. In the new derivation, the 1D equation is coupled to the 3D equation via a cylinder boundary source δ Γ , centered on the (2D) lateral boundary of the cylinder.…”

“…More recently, an extended finite element method was formulated for the mixed coupled 1D-3D flow model by Březina and Exner [2]. They take, as their starting point, the strong formulation of the coupled 1D-3D flow model (with a cylinder source [16]), before directly reformulating it as a mixed equation in 1D and 3D.…”

“…In this article, we derive a mixed formulation of the coupled 1D-3D flow model, taking the same set of model equations as in [16] as our starting point. Moreover, we follow their procedure of first deriving the variational formulation of the problem and then performing the averaging that leads to a dimensional reduction.…”

Well-Posedness and Discretization for a Class of Models for Mixed-Dimensional Problems with High-Dimensional Gap

“…Such models are referred to as mixed-dimensional when the network flow is simplified to a family of one-dimensional domains along with the network edges. 1 Moreover, when the coupling between the network and the domain exceeds two topological dimensions, the model is referred to as having a high-dimensional gap [21,19]. A high-dimensional gap thus arises when the flow in the network is connected to a domain of dimension d \geq 2 through its leaf nodes or when the flow in the network is connected to a domain of dimension d \geq 3 through its edges.…”