Projection method for quasiperiodic elliptic equations and application to quasiperiodic homogenization

TL;DR

This work develops an efficient numerical framework for elliptic PDEs with quasiperiodic coefficients by embedding the problem into a higher‑dimensional periodic domain via the Projection Method (PM). The PM yields a discrete, spectrally accurate (polynomial/spectral depending on data regularity) scheme solved through a highly scalable linear system $Q\\mathbf{U}=\\mathbf{F}$, where $Q=A\\circ W$ exhibits a multilevel block circulant structure. A compressed storage scheme together with a diagonal preconditioner produces the compressed PCG (C‑PCG) solver, delivering substantial memory savings and fast convergence for high‑dimensional problems. The approach avoids Diophantine approximation errors that plague periodic approximations and enables accurate computation of quasiperiodic homogenized coefficients, with extensive numerical validation across multiple frequency configurations. The results suggest broad applicability to quasiperiodic PDEs and lay groundwork for extensions to parabolic, wave, and nonlinear homogenization problems.

Abstract

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 polynomial 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.

Figures (6)

• Figure 1: One-dimensional quasiperiodic function $f=\cos(2\pi x)+\cos(2\pi\sqrt{2}x)$
• Figure 1: Diophantine approximation error of $\sqrt{2}$.
• Figure 2: The error $e_N$ against $N$ with $(\alpha_1, u_2)$ in PM, plotted on a log-log scale.
• Figure 3: The ratio $r=M_{\mathrm{PCG}}/M_\mathrm{C-PCG}$ vs. $N$ in log-log scale.
• Figure 4: Cost-Accuracy trade-off of PM and PAM $(L=1183, 1463)$ in solving \ref{['eqn:elliptic']} with $\alpha_2(x)$, respectively.

Theorems & Definitions (25)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Theorem 2.4
• Lemma 2.5: Equivalent modulus theorem braess2001finite
• Lemma 2.6: Poincaré inequality
• Proof 1
• Remark 3.1
• Theorem 3.1
• Proof 2