In this study, we 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, significantly reducing memory requirements. 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. Convergence analysis shows the spectral accuracy of the proposed method. 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：本研究针对具有拟周期系数的椭圆方程求解问题。为实现精确高效的计算，我们引入投影方法，将拟周期系统嵌入高维周期系统。为提升计算效率，我们基于刚度矩阵的多层块循环结构提出压缩存储策略，显著降低了内存需求。此外，我们设计了对角预条件子，通过降低刚度矩阵的条件数来高效求解所得高维线性系统。这些技术共同提升了所提方法的计算效能。收敛性分析表明该方法具有谱精度。通过一系列数值算例验证了所提方法的有效性与精确性。进一步地，我们将该方法应用于拟周期多尺度椭圆方程的均匀化系数高精度计算。