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 finite elements for the temporal and spatial discretization, 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 with the principal eigenvalue of the matrix $U^\top U$ as the basis of the Proper Orthogonal Decomposition (POD) method to obtain SEAM and a parallel SEAM. Numerical experiments confirm the efficiency of the new method.

翻译：本文针对具有齐次狄利克雷边界条件的二阶抛物型方程，提出了一种新型降基方法——单特征值加速法（SEAM）。高保真数值方法分别采用向后欧拉格式和协调有限元进行时间和空间离散。在时间步长足够小且时间步数不大的假设下，我们证明高保真解矩阵$U$的奇异值分布趋近于秩一矩阵的奇异值分布。通过选取矩阵$U^\top U$主特征值对应的特征函数作为本征正交分解（POD）方法的基，进而得到SEAM及并行SEAM。数值实验验证了新方法的有效性。