Mesh-Intrinsic GFEM: Asymptotic Smoothness on C^0 Unstructured Meshes

Abstract

MiGFEM constructs enriched local approximations from the original nodal degrees of freedom of a finite element mesh, without extra enrichment unknowns, thereby avoiding the linear dependence, ill-conditioning, energy inconsistency, and mass-lumping difficulties of conventional enriched formulations. This paper establishes asymptotic smoothness of MiGFEM on standard C0 unstructured meshes. Although the global approximation uses C0 partition-of-unity functions, its inter-element derivative jumps of order up to p decay as O(h^{p+1-|alpha|}) for solutions in C^{p+1}, and vanish identically when the exact solution lies in the local reconstruction space on every patch. The mechanism relies on interface coherence of patchwise local approximations and a derivative-space partition-of-unity identity (Partition of Zero) that cancels coherent interfacial traces. Asymptotic smoothness motivates the leading Leibniz derivative, which retains only the principal term of the full Leibniz expansion. This operator is globally continuous by construction; the neglected remainder shares the same asymptotic order as the inter-element jumps. Consequently, the same C0 MiGFEM trial space supports both weak Galerkin discretization via the full Leibniz derivative and strong-form collocation via the leading Leibniz derivative, on identical nodal unknowns. Numerical experiments verify the predicted jump decay and confirm exact cancellation under patchwise reproduction. The unified weak-strong framework is demonstrated on elasticity and biharmonic problems using unstructured meshes. A singular-enrichment test further shows that strong-form collocation reproduces enriched basis functions exactly without quadrature constraints.