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A New Reduced Basis Method for Parabolic Equations Based on Single-Eigenvalue Acceleration

A New Reduced Basis Method for Parabolic Equations Based on Single-Eigenvalue Acceleration

Year:    2024

Author:    Qijia Zhai, Qingguo Hong, Xiaoping Xie

Advances in Applied Mathematics and Mechanics, Vol. 16 (2024), Iss. 6 : pp. 1328–1357

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 $U^TU$ 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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.OA-2023-0053

Advances in Applied Mathematics and Mechanics, Vol. 16 (2024), Iss. 6 : pp. 1328–1357

Published online:    2024-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    30

Keywords:    Reduced basis method proper orthogonal decomposition singular value second order parabolic equation.

Author Details

Qijia Zhai

Qingguo Hong

Xiaoping Xie

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    https://doi.org/10.1016/j.jcp.2024.113481 [Citations: 0]