标题： 基于离散势和局部基函数的几何鲁棒非匹配边界代数方程方法 显示英文标题

标题： A geometrically robust unfitted boundary algebraic equation method based on discrete potentials and local basis functions

摘要： 我们提出了一种非匹配边界代数方程（BAE）方法，用于在复杂几何中求解椭圆偏微分方程。 该方法利用无限规则网格上的格点格林函数结合离散势理论来构造单层和双层势，这是边界积分方法的离散类比。 切割单元上的局部基函数能够适应任意边界条件，并与边界代数方程无缝集成。 差分势框架使得非齐次项的高效处理以及通过基于FFT的求解器快速计算层势成为可能。 我们通过一种新颖的插值算子框架建立了理论稳定性与收敛性。 所开发方法的关键优势包括：维度降低、几何灵活性、与网格无关的条件性、小切割稳定性以及对光滑和非光滑几何的统一处理。 数值实验验证了在不同纵横比和尖角的椭圆和菱形中的准确性和鲁棒性，并展示了在无限域中势流的应用。 显示英文摘要

摘要： We present an unfitted boundary algebraic equation (BAE) method for solving elliptic partial differential equations in complex geometries. The method employs lattice Green's functions on infinite regular grids combined with discrete potential theory to construct single and double layer potentials, which is a discrete analog to boundary integral method. Local basis functions on cut cells accommodate arbitrary boundary conditions and seamlessly integrate with the boundary algebraic equations. The difference potentials framework enables efficient treatment of nonhomogeneous terms and fast computation of layer potentials via FFT-based solvers. We establish theoretical stability and convergence through a novel interpolation operator framework. Key advantages of the developed method include: dimension reduction, geometric flexibility, mesh-independent conditioning, small-cut stability, and uniform treatment of smooth and non-smooth geometries. Numerical experiments validate accuracy and robustness across ellipses and diamonds with varying aspect ratios and sharp corners, and an application of potential flows in unbounded domains.