A mixed boundary value problem for the Laplace equation is considered in a rectangular domain. When using a grid method for the numerical solution of this problem, the error estimate typically involves the maximum moduli of derivatives of the sought solution, which makes such estimates difficult to apply in practice. In the literature, error estimates for certain methods expressed only in terms of the basic problem data are available. In particular, error bounds for Fourier-type discrete methods for the Dirichlet problem have been obtained in terms of known data, and related approaches based on majorant techniques and summability on layers have also led to estimates depending only on prescribed data. The discrete analogue of the Fourier method used in this work allows us to estimate the error of the method solely through the known data of the problem. This greatly simplifies the practical application of the proposed assessment.