TY - JOUR
T1 - A New Reduced Basis Method for Parabolic Equations Based on Single-Eigenvalue Acceleration
AU - Zhai , Qijia
AU - Hong , Qingguo
AU - Xie , Xiaoping
JO - Advances in Applied Mathematics and Mechanics
VL - 6
SP - 1328
EP - 1357
PY - 2024
DA - 2024/10
SN - 16
DO - http://doi.org/10.4208/aamm.OA-2023-0053
UR - https://global-sci.org/intro/article_detail/aamm/23470.html
KW - Reduced basis method, proper orthogonal decomposition, singular value, second order parabolic equation.
AB - <p style="text-align: justify;">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.</p>