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 an adaptive mesh refinement method numerically using 2-D steady incompressible lid-driven cavity flows and coarser meshes. The accuracy is shown by verifying that the centres of vortices given in the benchmarks are located in the refined grids of refined meshes for Reynolds numbers 100, 1000 and 2500 using coarser meshes.The adaptive mesh refinement method performs mesh refinement based on the numerical solutions of Navier'-Stokes equations solved by a finite volume method with a well known SIMPLE algorithm for pressure'velocity coupling. The accuracy of the refined meshes is shown by comparing the profiles of horizontal and vertical components of velocity fields with the corresponding components of the benchmarks together and drawing closed streamlines. The adaptive 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, steady state fluid flows or mass and heat transfer.