This paper presents four novel domain decomposition algorithms integrated with nonlinear mapping techniques to address collocation-based solutions of eigenvalue problems involving sharp interfaces or steep gradients. The proposed methods leverage the spectral accuracy of Chebyshev polynomials while overcoming limitations of existing tools like Chebfun, particularly in preserving higher-order derivative continuity and enabling flexible node clustering near discontinuities. Key findings include the following: for algorithm Performance: The one-point overlap method demonstrated significant improvements over global mapping approaches, reducing required grid points by orders of magnitude while maintaining spectral convergence. The two-point overlap method further methods generalized the approach, allowing arbitrary node distributions and nonlinear mappings. These achieved exponential error reduction for Burgers equation) by combining Taylor expansions with Chebyshev derivatives in overlap regions. While Chebfun splitting strategy automates domain decomposition, it enforces only C0 continuity, leading to discontinuous higher derivatives. In contrast, the proposed algorithms preserved smoothness up to CN continuity, critical for eigenvalue problems in hydrodynamic stability and nonlinear BVPs. Validation on 3D channel flow with viscosity stratification and Burgers equation highlighted the methods robustness. For instance, eigenvalue calculations for miscible core-annular flows matched prior results while resolving sharp viscosity gradients with fewer nodes.

翻译：本文提出了四种结合非线性映射技术的新型域分解算法，用于处理涉及尖锐界面或陡峭梯度的特征值问题的配点法求解。所提出的方法在利用切比雪夫多项式谱精度的同时，克服了现有工具（如Chebfun）的局限性，特别是在保持高阶导数连续性及在间断点附近实现灵活节点聚集方面。主要研究成果包括：在算法性能方面，单点重叠方法相较于全局映射方法展现出显著改进，在保持谱收敛的同时将所需网格点数量降低了数个数量级。两点重叠方法进一步推广了该思路，允许任意节点分布和非线性映射。这些方法通过在重叠区域结合泰勒展开与切比雪夫导数，实现了对Burgers方程误差的指数级衰减。尽管Chebfun的分割策略实现了域分解的自动化，但其仅强制C0连续性，导致高阶导数不连续。相比之下，所提出的算法保持了直至CN连续性的光滑度，这对于流体力学稳定性及非线性边值问题中的特征值求解至关重要。在具有黏度分层的三维通道流及Burgers方程上的验证凸显了该方法的鲁棒性。例如，可混溶核心-环状流的特征值计算在匹配已有结果的同时，以更少的节点解析了尖锐的黏度梯度。