A Hybrid Error Estimator for Finite Difference and Variational Iteration Methods via Operator Theory

This paper proposes a unified operator-theoretic framework for analyzing the Finite Difference Method (FDM) and the Variational Iteration Method (VIM) for solving nonlinear second-order boundary value problems (BVPs). The VIM iteration is interpreted as a Picard-type fixed-point iteration in a Banach space, and a computable a priori error bound is established under a Lipschitz condition on the nonlinearity. Concurrently, FDM is treated as a finite-rank projection method, and its convergence is demonstrated via consistency and stability. The main contribution of this work is a novel hybrid a posteriori error estimator that uses the continuous VIM iterate to bound the global discretization error of the FDM solution without requiring additional PDE solves or mesh-doubling, requiring only the VIM iteration already computed as part of the error estimation process. Unlike traditional residual-based estimators, the proposed estimator relies solely on the discrepancy between the smooth semi-analytical approximation and the discrete numerical solution. Theoretical convergence rates and the efficiency of the estimator are validated through numerical experiments on a nonlinear BVP with a known exact solution and a non-solvable linear BVP. The results indicate that the hybrid estimator provides a reliable and computationally efficient indicator for adaptive mesh refinement.