In this study, our main objective is to address the challenge of solving elliptic equations with quasiperiodic coefficients. To achieve accurate and efficient computation, we introduce the projection method, which enables the embedding of quasiperiodic systems into higher-dimensional periodic systems. To enhance the computational efficiency, we propose a compressed storage strategy for the stiffness matrix by its multi-level block circulant structure, reducing memory requirements while preserving accuracy. Furthermore, we design a diagonal preconditioner to efficiently solve the resulting high-dimensional linear system by reducing the condition number of the stiffness matrix. These techniques collectively contribute to the computational effectiveness of our proposed approach. We demonstrate the effectiveness and accuracy of our approach through a series of numerical examples. Moreover, we apply our method to achieve a highly accurate computation of the homogenized coefficients for a quasiperiodic multiscale elliptic equation.

翻译：本研究的主要目标是解决具有准周期系数的椭圆方程求解难题。为实现精确高效的计算，我们引入了投影方法，该方法可将准周期系统嵌入到更高维的周期系统中。为提升计算效率，我们提出了一种基于刚度矩阵多级块循环结构的压缩存储策略，在保持精度的同时降低内存需求。此外，我们设计了一种对角预处理器，通过降低刚度矩阵的条件数来高效求解高维线性系统。这些技术共同提升了所提方法的计算效能。通过一系列数值算例验证了该方法的有效性与精确性，并进一步将其应用于准周期多尺度椭圆方程均匀化系数的高精度求解。