Patch-Local Coupled Quadrature for Galerkin RBF-PUM : Construction, Assembly, and Robustness

Sammanfattning

Galerkin radial basis function partition of unity methods provide a flexible meshfree framework for solving partial differential equations, but their weak-form discretization requires accurate quadrature for non-polynomial basis functions and overlapping patch contributions. Global quadrature constructions can be accurate, but they reduce the practical locality of the method. This work investigates whether quadrature weights can instead be constructed locally on patches and then assembled into an accurate global weak-form discretization. The proposed approach computes interior and boundary quadrature weights together using compatibility conditions derived from Green’s identities. In the patch-local setting, the construction is applied on cut domains formed by the intersection between the physical domain and each patch. Artificial interfae-boundary weights are included to close the loacl compatibility systems, while only physical-boundary weights are assembled into the global Poisson boundary terms. The resulting patch-local weights are assembled globally using either direct owner-based assembly or partition-of-unity-weighted blending. A global coupled quadrature construction and a single-domain formal analogue are used as baselines to separate the effect of the coupled quadrature principle from the effects of patch localization and global assembly. Numerical experiments for a manufactured Poisson problem show that the global coupled baseline gives the most accurate and consistent results, with the smallest PDE errors and moment errors. The single-domain formal analogue reproduces the global baseline, indicating that the coupled interior-boundary quadrature principle is effective when imposed globally. The patch-local construction remains numerically usable and gives reasonable PDE errors, and blended assembly generally improves over direct assembly. However, the local methods do not fully recover the global-baseline accuracy. Additional diagnostics show no simple local rank collapse or physical-boundary weight blow-up, but reveal conditioning sensitivity, especially for interior patches. The results suggest that the main difficulty is not the coupled quadrature itself, but the preservation of patch-local consistency through localization and global assembly