Abstract
In this paper, we develop a new reduced basis (RB) method, named as Single Eigenvalue Acceleration Method (SEAM), for second order parabolic equations with homogeneous Dirichlet boundary conditions. The high-fidelity numerical method adopts the backward Euler scheme and conforming simplicial finite elements for the temporal and spatial discretizations, respectively. Under the assumption that the time step size is sufficiently small, and time steps are not very large, we show that the singular value distribution of the high-fidelity solution matrix U is close to that of a rank one matrix. We select the eigenfunction associated to the principal eigenvalue of the matrix U>U as the basis of Proper Orthogonal Decomposition (POD) method so as to obtain SEAM and a parallel SEAM. Numerical experiments confirm the efficiency of the new method.
Recommended Citation
Q. Zhai et al., "A New Reduced Basis Method for Parabolic Equations based on Single-Eigenvalue Acceleration," Advances in Applied Mathematics and Mechanics, vol. 16, no. 6, pp. 1328 - 1357, Global Science Press, Dec 2024.
The definitive version is available at https://doi.org/10.4208/aamm.OA-2023-0053
Department(s)
Mathematics and Statistics
Keywords and Phrases
proper orthogonal decomposition; Reduced basis method; second order parabolic equation; singular value
International Standard Serial Number (ISSN)
2075-1354; 2070-0733
Document Type
Article - Journal
Document Version
Citation
File Type
text
Language(s)
English
Rights
© 2024 Global Science Press, All rights reserved.
Publication Date
01 Dec 2024
Comments
King Abdullah University of Science and Technology, Grant 12171340