Residual-Based Adaptive Refinement in Hybrid Variational-Collocation PINNs for Singular Elliptic PDEs with Mixed Dirichlet-Neumann Boundary Conditions

Solutions of second-order elliptic PDE problems on nonconvex or cracked domains with mixed Dirichlet-Neumann conditions typically lie outside $H^2(Ω)$. Near re-entrant corners, crack tips and points where the boundary condition changes type, the gradient grows like $r^{λ-1}$ with $λ<1$. We study res