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

• Published in: Mathematics (Basel) Vol. 11; no. 16; p. 3592
• Main Authors: Amosov, Andrey, Gómez, Delfina, Panasenko, Grigory, Pérez-Martinez, Maria-Eugenia
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01-08-2023; MDPI
• Subjects: Approximation; approximation of the spectrum; Approximation theory; asymptotic domain decomposition method; Asymptotic methods; Asymptotic properties; Boundary conditions; Continuity; Decomposition (Mathematics); diffusion operator; Domain decomposition methods; Eigenvalues; Eigenvectors; junction conditions; Mathematical research; Mathematics; thin tubes; Tubes; United States; 35P05; 74K30; diffusion operator; approximation of the spectrum; thin tubes; junction conditions MSC: 35J05; asymptotic domain decomposition method; 35P10
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math11163592

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.