Numerical solutions to the classical Graetz problem in the entrance region of a pipe produce estimates of the Nusselt number which are greatly in error except when the uniform grid is extremely fine, leading to huge storage requirements. The use of adaptive grids, it is proposed, will resolve the twin problem of excessive storage space and inaccuracy of results. Therefore, two adaptive grids using different methods of generating the weighting function are developed and tested in this work. One of the adaptive grids produced results that compared favourably with the exact solution with as few as 11 grid points. With the uniform grid, as many as 301 grid points would have been required.