The Finite Element Method is the method of choice for the solution of elliptic boundary value problems. In computing these approximations we follow two aims: We need a computable error bound to judge the quality of an approximation, and we want to reduce the amount of work to obtain an approximation of a prescribed tolerance. The first aim is a must since numerical simulations without information about their accuracy may be not reliable. This has been seen by some failures in the past, see information about the Sleipner accident. The second aim may be achieved by constructing local (e.g.\ in space) error indicators and perform local refinement where large errors are indicated. We show for a model problem how to construct convergent local refinement algorithms and show that the algorithm has optimal complexity.