Dynamic mesh adaptation is a very useful technique for reducing the computational time and memory requirements when solving evolutionary partial differential equations. The reduction is greater when the solution exhibits localized behaviour as in the case of a moving front where the 'action' occurs over a small fraction of the domain. Difficulties arising in the use of dynamic grid adaptation include significant overhead, added storage, and errors introduced due to grid manipulation. We propose a method that maintains a fine ungoform grid in the important regions of the domain by using inexpensive action indicators to trigger selective refinement. The method is simple to code and adapt to existing finite element solvers. It requires low added storage and overhead per element, and can significantly reduce grid manipulation errors. We present the selective refinement method and its use of the solution in the gridding decision process, and detail the streamlined storage structure. Theoretical speedup compared to fixed grid methods is derived and the improvement in storage is analysed. Finally, we use the method to solve some problems exhibiting localized behaviour in one dimension and compare to theory.