Approximation of eigenvalues and eigenfunctions of the diffusion operator in a domain containing thin tubes by asymptotic domain decomposition method

Abstract

We consider the spectral problem for the diffusion operator considered in a domain containing thin tubes. A new version of the method of partial asymptotic decomposition of the domain is introduced to reduce the dimension inside the tubes, getting a model of hybrid dimensions. The method truncates the tubes at some small distance from the ends of the tubes and replaces the longer part of the tubes with segments. At the interface of the three-dimensional and one-dimensional subdomains, special junction conditions are set: the pointwise continuity of the flux and the continuity of the average over a cross-section of the eigenfunctions. We obtain conditions on the ratio of the characteristic sizes in the transverse and longitudinal directions that ensure the closeness of two spectra, i.e. of the diffusion operator in the full-dimensional domain and the partially reduced one, keeping the conservation of the multiplicity, all up to a prescribed accuracy.

Keywords:

• Asymptotic domain decomposition method
• spectral perturbations
• diffusion operator
• thin tubes
• junction conditions

2020 Mathematics Subject Classifications:

• 35B40
• 35P15
• 35P20

Acknowledgments

The article was written in equal co-authorship. All authors have read and agreed to the published version of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The results of the first author were obtained as part of the implementation of the state assignments of the Ministry of Education and Science of Russia (project FSWF-2023-0012), second and fourth authors were supported by Ministerio de Ciencia e Innovación grant by the grant PID2022-137694NB-I00 funded by MICIU/AEI/10.13039/501100011033 and by ERDF/EU.