We investigate the behaviour of two efficiency based strategies for adaptive grid refinement. We assume that multigrid solvers are used such that the work for solving the linear systems is proportional to the number of unknowns. Refinement decisions are based on maximizing the efficiency, that is, for a given error bound, our objective is to find a mesh sequence minimizing the total work. To achieve this, we want to use a measure that will tell us how to proceed from the current refinement level to the next level. Two strategies are presented: one is minimizing work times error, and the other is minimizing the effective error reduction. We consider an exact solution that has a singularity. We first consider h-refinement. It is observed that both strategies result in a highly accurate mesh sequence efficiently. However, they fail for highly singular solutions. A modification is given for this situation by taking graded mesh refinement for the element which contains a singularity. It is shown that this modification improves the performance of both strategies significantly. The efficiency of these modified strategies is also shown by comparing with a threshold-based strategy. At last, we give similar strategies for hp-refinement.