Finite difference methods remain a cornerstone in computational science due to their simplicity and computational efficiency. However, deriving a posteriori error estimates for these methods remains a significant challenge. First-order finite difference schemes can be interpreted as finite element (FE) schemes, allowing the use of residual-based error estimators. However, the literature has fewer contributions for higher-order finite difference schemes. In this work, we propose a recovery-based error indicator for higher-order finite difference discretizations of the Poisson problem and the Wave equation. Specifically, we construct a solution in a finite element space using the finite difference solution and then apply a recovery-based error indicator, introduced by A. Naga and Z. Zhang, to estimate the gradient of the error of this constructed solution. We demonstrate the accuracy of the proposed approach through several numerical experiments.