Lid-driven cavity flows have been widely investigated and accurate results have been achieved as benchmarks for testing the accuracy of computational methods. This paper verifies the accuracy of a mesh refinement method numerically using two-dimensional steady incompressible lid-driven flows and finer meshes. The accuracy is shown by comparing the coordinates of centres of vortices located by the mesh refinement method with the corresponding benchmark results. The accuracy verification shows that the mesh refinement method provides refined meshes that all centres of vortices are contained in refined grids based on the numerical solutions of Navier-Stokes equations solved by finite volume method except for one case. The well known SIMPLE algorithm is employed for pressure-velocity coupling. The accuracy of the numerical solutions is shown by comparing the profiles of horizontal and vertical components of velocity fields with the corresponding components of the benchmarks and also streamlines. The mesh refinement method verified in this paper can be applied to find the accurate numerical solutions of any mathematical models containing continuity equations for incompressible fluid or steady state fluid flows or heat transfer.