非拟合网格上椭圆界面问题的局部杂交有限元-随机特征方法
A Localized Hybrid Finite Element--Random Feature Method for Elliptic Interface Problems on Unfitted Meshes
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- Ministry of Education Key Laboratory for NSLSCS（教育部非线性科学重点实验室）
- School of Mathematical Sciences, Nanjing Normal University（南京师范大学数学科学学院）
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中文总结 AI 辅助
针对非拟合网格上的椭圆界面问题，提出仅在界面切割单元构成的窄带内采用随机特征方法、其余单元用标准有限元的局部杂交方法，通过残差方程处理界面条件并耦合有限元与随机特征分量，数值验证了其有效性。
中文摘要 AI 辅助
我们针对非拟合背景网格上的椭圆界面问题，提出了一种局部杂交有限元-随机特征方法（FEM-RFM）。核心思路是仅在均匀笛卡尔背景网格的界面切割单元构成的窄带内使用随机特征方法（RFM），而在其余未切割单元上保留标准$Q_p$有限元。在该窄带内，界面的两个物理侧采用局部RFM表示；通过涉及物理界面上两个RFM表示的残差方程，引入解和法向通量的规定跳跃$\Gamma$，同时有限元方法（FEM）与RFM分量通过迹和变分法向通量残差，在另一个与网格对齐的人工界面$\Gamma_c$上耦合。舒尔约化通过稀疏求解消除内部有限元未知量，同时保留其伽辽金方程作为等式约束；剩余的窄带和界面残差在仅包含耦合节点值与随机特征系数的约化最小二乘问题中最小化。通过数值方法对一系列二维和三维界面问题评估该方法，这些问题涉及高系数对比度、非光滑与多重界面、不规则外边界，以及面向应用的周期双连续复合材料热均匀化问题。
英文摘要
We develop a localized hybrid finite element--random feature method (FEM--RFM) for elliptic interface problems on unfitted background meshes. The central idea is to use the random feature method (RFM) only in the narrow band formed by interface-cut cells of a uniform Cartesian background mesh,while retaining standard $Q_p$ finite elements on the remaining uncut cells. Within the band, local RFM representations are used on the two physical sides of the interface. The prescribed jumps in the solution and normal flux are incorporated through residual equations involving the two RFM representations on the physical interface $Γ$, while the finite element method (FEM) and RFM components are coupled across a separate grid-aligned artificial interface $Γ_c$ through trace and variational normal-flux residuals. A Schur reduction eliminates the interior finite element unknowns through sparse solves while retaining their Galerkin equations as equality constraints. The remaining band and interface residuals are minimized in a reduced least-squares problem involving only the coupling nodal values and random-feature coefficients. The method is assessed numerically on a range of two- and three-dimensional interface problems involving high coefficient contrasts, nonsmooth and multiple interfaces, and irregular outer boundaries, together with an application-oriented thermal-homogenization problem for a periodic bicontinuous composite.
