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

Abstract

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 \[ X_θ=(\mathcal E_p,\mathrm{Dom}(C))_{θ,1} \] and show that global convergence can be recovered by working on the higher regularity scale \[ Y_θ=(\mathrm{Dom}(C),\mathrm{Dom}(C^2))_{θ,1}. \] Under suitable stability assumptions, finite-time convergence estimates of order \(O(h^θ)\) 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.