A New Reduced Basis Method for Parabolic Equations Based on Single-Eigenvalue Acceleration (Advances in Applied Mathematics and Mechanics, 2024)
In this paper, we develop a new reduced basis (RB) method, named as SingleEigenvalue Acceleration Method (SEAM), for second order parabolic equations withhomogeneous Dirichlet boundary conditions. The high-fidelity numerical methodadopts the backward Euler scheme and conforming simplicial finite elements for thetemporal and spatial discretizations, respectively. Under the assumption that the timestep 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 rankone matrix. We select the eigenfunction associated to the principal eigenvalue of thematrix $U^TU$ as the basis of Proper Orthogonal Decomposition (POD) method so as toobtain SEAM and a parallel SEAM. Numerical experiments confirm the efficiency ofthe new method.