Asymptotic Domain Decomposition Method for Approximation the Spectrum of the Diffusion Operator in a Domain Containing Thin Tubes

Andrey Amosov (), Delfina Gómez, Grigory Panasenko and Maria-Eugenia Pérez-Martinez
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Andrey Amosov: Department of Mathematical and Computer Modelling, National Research University “Moscow Power Engineering Institute”, Krasnokazarmennay St. 14, 111250 Moscow, Russia
Delfina Gómez: Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Av. de los Castros s/n., 39005 Santander, Spain
Grigory Panasenko: Institute of Applied Mathematics, Vilnius University, Naugarduko Str., 24, 03225 Vilnius, Lithuania
Maria-Eugenia Pérez-Martinez: Departamento Mathematica Aplicada y Ciencias de la Computación, Universidad de Cantabria, Av. de los Castros s/n., 39005 Santander, Spain

Mathematics, 2023, vol. 11, issue 16, 1-25

Abstract: The spectral problem for the diffusion operator is 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. It truncates the tubes at some small distance from the ends of the tubes and replaces 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. The existence of the discrete spectrum is proved for this partially reduced problem of the hybrid dimension. The conditions of the closeness of two spectra, i.e., of the diffusion operator in the full-dimensional domain and the partially reduced one, are obtained.

Keywords: asymptotic domain decomposition method; approximation of the spectrum; diffusion operator; thin tubes; junction conditions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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