A mesh adaptation scheme is presented that employs edge collapse to coarsen the mesh and selective point placement combined with an incremental Delaunay algorithm for mesh enrichment. The coarsening and enrichment procedures can be used in isolation to achieve a solution adapted mesh for time varying computations whose boundaries remain fixed. These procedures can also be combined with a mesh movement technique to form an adaptation algorithm that will modify the mesh of a domain whose shape is evolving in time. The adaptation techniques are described and examples are presented to illustrate the method for time dependent problems arising from crack nucleation studies and vortex shedding of incompressible flow over a cylinder. (C) 2003 Kluwer Academic Publishers.