发表机构
Shanghai Normal University; Nanyang Technological University（上海师范大学; 南洋理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对多维二阶椭圆型问题配点法的长期病态性，提出基于 Birkhoff 插值的良态配点法，通过稳定对角化构造高效预条件子，显著降低条件数并实现几乎与阶数无关的 GMRES 收敛。
AI 中文摘要
基于 Birkhoff 插值在 Gauss 型点上的配点法对一维初值和边值问题是良态的 [Wang 等, {\em SIAM J. Sci. Comput.} 36 (2014)]，但其底层构造不能直接推广到多维情形。我们解决了多维配点法对二阶椭圆型问题长期存在的病态性问题。关键观察是：由 Legendre-Gauss-Lobatto 点上的 Birkhoff 插值构造的二阶微分矩阵及其逆——伪谱积分矩阵 (PSIM)——都与对称负定矩阵相似。这使得稠密、非对称且病态的微分和积分矩阵能够被稳定地对角化，即使配点数量达数千个也如此，并由此构造出高效的多维 Birkhoff 预条件子。对于变系数问题，系数被直接纳入对角化和预条件子构造中，这对高度各向异性、高对比度、振荡和退化椭圆算子至关重要。我们提供了谱分析以及广泛的二维和三维数值实验，展示了条件数的显著降低和几乎与多项式阶数无关的 GMRES 收敛性，同时保持高阶精度。所得到的 Birkhoff 配点格式使多维谱配点法对具有挑战性的椭圆问题变得实用。
英文摘要
Collocation methods based on Birkhoff interpolation at Gaussian-type points are well-conditioned for one-dimensional initial and boundary value problems [Wang et al., {\em SIAM J. Sci. Comput.} 36 (2014)], but the underlying construction does not extend directly to multiple dimensions. We address the long-standing ill-conditioning of multidimensional collocation methods for second-order elliptic-type problems. The key observation is that the second-order differentiation matrix and its inverse, the pseudospectral integration matrix (PSIM) constructed from Birkhoff interpolation at Legendre-Gauss-Lobatto points, are both similar to symmetric negative definite matrices. This enables stable diagonalisation of the dense, non-symmetric and ill-conditioned differentiation and integration matrices, even for thousands of collocation points, and leads to efficient multidimensional Birkhoff preconditioners. For variable-coefficient problems, the coefficients are incorporated directly into the diagonalisation and preconditioner construction, which is essential for highly anisotropic, high-contrast, oscillatory and degenerate elliptic operators. We provide spectral analysis and extensive two- and three-dimensional numerical experiments, demonstrating substantial reductions in condition numbers and nearly polynomial-degree-independent GMRES convergence while retaining high-order accuracy. The resulting Birkhoff-collocation schemes make multidimensional spectral collocation methods practical for challenging elliptic problems.
Comments36 pages, 11 figures
