发表机构
College of Science, National University of Defense Technology; School of Mathematics and Statistics, HNP-LAMA, Central South University; Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University（国防科技大学理学院; 中南大学数学与统计学院; 南洋理工大学物理与数学科学学院数学科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对变系数双调和方程，提出一种增广高阶紧致有限差分法，通过将不可用边界值作为未知数降低耦合，结合无矩阵GMRES和FFT快速求解器，实现四阶精度和$O(n\log n)$复杂度，可高效求解大规模问题。
AI 中文摘要
我们针对具有夹紧边界条件和变系数的双调和方程，提出了一种增广高阶紧致有限差分方法。标准的混合型公式引入了一个辅助变量，但其边界值不可用，导致所得离散系统全局耦合，难以在大规模下求解。我们的关键贡献是开发了一种新的增广公式，将这些不可用的边界值视为额外未知数，将全局耦合降低为低维Schur补系统，并产生解耦的二阶子问题。Schur补通过无矩阵GMRES求解，而子问题由基于FFT的快速求解器处理。该方法使用紧致模板实现四阶精度，计算复杂度为$O(n\log n)$，使得在几分钟内求解具有$1024^3$自由度的双调和方程成为可能。据我们所知，这种计算效率水平此前在文献或实践中均未达到。利用能量估计和傅里叶分析，我们推导了具有不精确Dirichlet边界的Poisson方程的新$L^2$估计，然后证明了所提出格式的收敛性。我们提供了大量的数值实验来确认精度和效率，并将该快速且精确的求解器进一步应用于三调和方程、高波数问题、Stokes流和板弯曲问题。
英文摘要
We propose an augmented high-order compact finite difference method for biharmonic equations with clamped boundary conditions and variable coefficients. Standard mixed-type formulations introduce an auxiliary variable, but its boundary values are unavailable, leaving the resulting discrete systems globally coupled and difficult to solve at large scales. Our key contribution is the development of a new augmented formulation that treats these unavailable boundary values as additional unknowns, reduces the global coupling to a lower-dimensional Schur complement system, and yields decoupled second-order subproblems. The Schur complement is solved by matrix-free GMRES, while the subproblems are handled by FFT-based fast solvers. The method achieves fourth-order accuracy using compact stencils, and has $O(n\log n)$ computational complexity, enabling the solution of the biharmonic equation with $1024^3$ degrees of freedom within several minutes. To the best of our knowledge, this level of computational efficiency has not previously been achieved in either the literature or practice. Using energy estimates and Fourier analysis, we derive a new $L^2$-estimate for Poisson equations with inexact Dirichlet boundary and then prove the convergence of the proposed scheme. We provide ample numerical experiments to confirm the accuracy, efficiency, and further apply the fast and accurate solver to triharmonic equations, high-wavenumber problems, Stokes flow, and plate bending problems.
Comments20 pages, 6 figures
