核心概念
This study presents a highly efficient algorithm for solving quasiperiodic elliptic equations and computing homogenized coefficients for quasiperiodic multiscale problems. The key innovations include the use of the projection method to transform the quasiperiodic problem into a higher-dimensional periodic system, a compressed storage format for the stiffness matrix, and a diagonal preconditioner to accelerate the iterative solver.
摘要
The content starts by introducing the concept of quasiperiodic functions and quasiperiodic Sobolev spaces, which are essential for establishing the well-posedness of quasiperiodic elliptic equations.
The authors then present the projection method (PM) as an accurate and efficient approach for solving quasiperiodic systems. PM involves transforming the quasiperiodic function into a high-dimensional periodic function, allowing the use of efficient Fourier-based techniques. The authors derive the discrete scheme and linear system for the quasiperiodic elliptic equation using the PM.
To address the challenges of the large linear system, the authors introduce a compressed storage method for the stiffness matrix, which leverages the multilevel block circulant structure to significantly reduce memory requirements. Additionally, they propose a diagonal preconditioner to accelerate the convergence of the preconditioned conjugate gradient (PCG) method, resulting in the compressed PCG (C-PCG) algorithm.
The authors present numerical experiments to validate the accuracy and efficiency of their proposed approach. They compare the performance of PM against the periodic approximation method (PAM) in solving quasiperiodic elliptic equations, demonstrating the superior accuracy and robustness of PM. Furthermore, they apply their method to compute the homogenized coefficients for a quasiperiodic multiscale elliptic equation, achieving highly accurate results.
統計資料
The authors provide several numerical examples to demonstrate the effectiveness of their proposed algorithm. In the first example, the spectral points of the elliptic coefficient and the solution are two separated incommensurate frequencies, 1 and √2. The authors compare the CPU time and memory usage between standard PCG and C-PCG, showing that C-PCG can significantly reduce the computational time and memory requirements.
The authors also compare the accuracy and efficiency of PM against PAM in solving the quasiperiodic elliptic equation. They observe that the numerical error of PAM is mainly controlled by the Diophantine approximation error, and PAM fails to achieve improved accuracy even with a significantly higher number of discrete points. In contrast, PM exhibits superior accuracy and robustness.
引述
"PM utilizes the spectral collocation method by employing the discrete Fourier-Bohr transformation to approximate quasiperiodic functions. By efficiently computing higher-dimensional periodic parent functions and incorporating a projection matrix, accurate approximations of quasiperiodic functions can be obtained."
"To tackle the difficulties associated with the large linear system, we introduce a fast algorithm that utilizes a compressed storage format and a diagonal preconditioner. By employing the compressed storage format, we can effectively store and manipulate the tensor product matrices involved in the linear system. This approach enables efficient handling of the memory-intensive matrices arising from the quasiperiodic problem."