A Discrete Resolvent Framework for Delay Differential Equations: Local Defects, Global Propagation, and Splitting Approximations

We develop a discrete resolvent framework for implicit Euler and Lie–Trotter splitting approximations of delay differential equations. The analysis is formulated entirely in terms of discrete propagators acting on product spaces and does not rely on semigroup generation or evolution-family theory. We establish local defect estimates on fractional interpolation spaces $$X_{\theta }=({\mathcal{E}}_p,\mathrm{D}\mathrm{o}\mathrm{m}(C){)}_{\theta ,1}$$ and show that global convergence can be recovered by working on the higher regularity scale $$Y_{\theta }=(\mathrm{D}\mathrm{o}\mathrm{m}(C),\mathrm{D}\mathrm{o}\mathrm{m}(C^2){)}_{\theta ,1}\mathrm{.}$$ Under suitable stability assumptions, finite-time convergence estimates of order $O(h^{\theta })$ are obtained for both autonomous and non-autonomous problems. The framework further applies to sectorial block models of reaction–diffusion type. Numerical experiments support the theoretical results.