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Poisson-type problems with transmission conditions at boundaries of infinite metric trees

Maryna Kachanovska, Kiyan Naderi, Konstantin Pankrashkin

TL;DR

The paper establishes a rigorous framework for Poisson-type problems on a hybrid structure combining a Euclidean exterior domain $\Omega$ with an infinite metric tree $\mathcal{T}$ glued along $\Gamma$. It introduces embedded trace maps to glue the two components and reduces solvability to a boundary-integral equation involving the tree and exterior Dirichlet-to-Neumann maps, proving well-posedness under mild conditions. Finite-tree truncations and a condensation technique yield efficient and provably convergent approximations, with explicit rates depending on regularity; the framework is adaptable to applications such as fractal/tree antennas. The results advance both the theoretical understanding of mixed-dimensional transmission problems and their numerical treatment via stable, scalable DTN-based formulations.

Abstract

The paper introduces a Poisson-type problem on a mixed-dimensional structure combining a Euclidean domain and a lower-dimensional self-similar component touching a compact surface (interface). The lower-dimensional piece is a so-called infinite metric tree (one-dimensional branching structure), and the key ingredient of the study is a rigorous definition of the gluing conditions between the two components. These constructions are based on the recent concept of embedded trace maps and some abstract machineries derived from a suitable Green-type formula. The problem is then reduced to the study of Fredholm properties of a linear combination of Dirichlet-to-Neumann maps for the tree and the Euclidean domain, which yields desired existence and uniqueness results. One also shows that finite sections of tree can be used for an efficient approximation of the solutions.

Poisson-type problems with transmission conditions at boundaries of infinite metric trees

TL;DR

The paper establishes a rigorous framework for Poisson-type problems on a hybrid structure combining a Euclidean exterior domain with an infinite metric tree glued along . It introduces embedded trace maps to glue the two components and reduces solvability to a boundary-integral equation involving the tree and exterior Dirichlet-to-Neumann maps, proving well-posedness under mild conditions. Finite-tree truncations and a condensation technique yield efficient and provably convergent approximations, with explicit rates depending on regularity; the framework is adaptable to applications such as fractal/tree antennas. The results advance both the theoretical understanding of mixed-dimensional transmission problems and their numerical treatment via stable, scalable DTN-based formulations.

Abstract

The paper introduces a Poisson-type problem on a mixed-dimensional structure combining a Euclidean domain and a lower-dimensional self-similar component touching a compact surface (interface). The lower-dimensional piece is a so-called infinite metric tree (one-dimensional branching structure), and the key ingredient of the study is a rigorous definition of the gluing conditions between the two components. These constructions are based on the recent concept of embedded trace maps and some abstract machineries derived from a suitable Green-type formula. The problem is then reduced to the study of Fredholm properties of a linear combination of Dirichlet-to-Neumann maps for the tree and the Euclidean domain, which yields desired existence and uniqueness results. One also shows that finite sections of tree can be used for an efficient approximation of the solutions.

Paper Structure

This paper contains 15 sections, 30 theorems, 262 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Boundary value problems on the metric tree
  3. Function spaces
  4. Laplacian
  5. Multi-scale boundary decomposition and trace map
  6. Normal derivative and Dirichlet-to-Neumann map
  7. Boundary values on finite sections
  8. Approximations of the Dirichlet-to-Neumann map
  9. Boundary value problems on the exterior domain
  10. Surface potentials
  11. Dirichlet-to-Neumann map for $m\ge 3$
  12. Dirichlet-to-Neumann map for $m=2$
  13. Hybrid transmission problem
  14. Well-posedness
  15. Approximations by finite truncations

Key Result

Lemma 2.1

The embedding $H^1(\mathcal{T})\hookrightarrow L^2(\mathcal{T})$ is compact, and there is a constant $C_0>0$ such that and is a scalar product on $\widetilde{H}^1(\mathcal{T})$ which is equivalent to the induced scalar product inherited from $H^1(\mathcal{T})$.

Figures (4)

  • Figure 1: An illustration of the structure of the tree $\mathcal{T}$ with $p=3$.
  • Figure 2: Gluing between the tree $\mathcal{T}$ and the exterior domain $\Omega$.
  • Figure 3: The tree $\mathcal{T}$
  • Figure 4: The subtrees $\mathcal{T}_{n,k}$ and $\mathcal{T}^j_{n,k}$.

Theorems & Definitions (61)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • Definition 2.6: Multiscale decomposition, Euclidean case
  • Definition 2.7: Multiscale decomposition, manifold case
  • ...and 51 more