2026/08/03 by Marin Mišur
Computer Science · Mathematics · #cs.NA #math.NA
Work in progress
arxiv created 2026/08/04 · arxiv updated 2026/08/05
The regularity of solutions to elliptic partial differential equations degrades severely at mixed Dirichlet-Neumann boundary junctions, characterized classically by an O(r1/2) leading singular function. While this linear behavior is well documented, the introduction of gradient-dependent semilinear perturbations alters the local asymptotic profile. This article proves that gradient penalties scaling linearly with |∇ u| induce a highly localized O(r-1/2) source term that resonates with the half-integer spectrum of the principal homogeneous differential operator. This non-orthogonal resonance causes standard separable polynomial assumptions to fail at order O(r3/2). We establish a generalized resonance theorem that forces the emergence of a logarithmic anomaly, providing the exact analytical formulation of the resulting r3/2 ln(r) profile alongside rigorous local Sobolev regularity bounds for the remainder. By calculating the exact logarithmic coefficients and angular offsets for ℓ1, ℓ2, ℓ_∞, and arbitrary ℓq-norm penalties, we demonstrate the universality of this obstruction. Finally, we formalize the corresponding enriched continuous Galerkin space (XFEM), establish its quasi-optimality, and present finite element experiments that confirm the predicted localized pollution, recover the predicted logarithmic coefficients across the ℓ1, ℓ2, and ℓ_∞ penalties on a curvature-free flat junction, and show that resolving the junction recovers optimal degree-of-freedom efficiency in standard finite element solvers; we close by outlining the targeted software architectures required for a fully enriched implementation.