Solution verification is crucial for establishing the reliability of simulations. A central challenge is to estimate the discretization error accurately and reliably. Many approaches to this estimation are based on the observed order of accuracy; however, it may fail when the numerical solutions lie outside the asymptotic range. Here we propose a grid refinement method that adopts constant orders given by the user, called the prescribed orders expansion method (POEM). Through an iterative procedure, the user is guaranteed to obtain the dominant orders of the discretization error. The user can also compare the corresponding terms to quantify the degree of asymptotic convergence of the numerical solutions. These features ensure that the estimation of the discretization error is accurate and reliable. Moreover, the implementation of POEM is the same for any dimensions and refinement paths. We demonstrate these capabilities using some advection and diffusion problems and standard refinement paths. The computational cost of using POEM is lower if the refinement ratio is larger; however, the number of shared grid points where POEM applies also decreases, causing greater uncertainty in the global estimates of the discretization error. We find that the proportion of shared grid points is maximized when the refinement ratios are in a certain form of fractions. Furthermore, we develop the method of interpolating differences between approximate solutions (MIDAS) for creating shared grid points in the domain. These approaches allow users of POEM to obtain a global estimate of the discretization error of lower uncertainty at a reduced computational cost.