This paper addresses the problem of simultaneous localisation and mapping (SLAM) by a mobile robot. An incremental SLAM algorithm is introduced that is derived from multigrid methods used for solving partial differential equations. The approach improves on the performance of previous relaxation methods for robot mapping because it optimizes the map at multiple levels of resolution. The resulting algorithm has an update time that is linear in the number of estimated features for typical indoor environments, even when closing very large loops, and offers advantages in handling non-linearities compared to other SLAM algorithms. Experimental comparisons with alternative algorithms using two well-known data sets and mapping results on a real robot are also presented