Local adaptation is necessary to achieve accurate solutions with
an acceptable effort in terms of simulation time and memory consumption.
The local refinement, coarsening, or smoothing steps are performed to
enhance purely geometrical quality aspects as discussed in
Section 3.1, or are guided by a control function as explained
in the last section. In the first case the refined regions concentrate
around areas where the local feature size is small. Large elements which
resolve small geometrical features are usually badly shaped and require
refinement.
In the latter case the mesh density is adapted to a stationary
solution or dynamically for each timestep of a transient simulation.
The regions of refinement have to migrate as the characteristics of
the transient solution change over the domain.
Essentially, local refining in some regions as well as local
coarsening in other regions becomes necessary to avoid meshing the
entire domain repeatedly.
The here discussed refinement is often referred to as h-refinement
which results in a decrease of the stepsize 
.
On the other hand are p-refinement techniques which increase
the order of the polynomial form functions of the finite element
approximation.
Topics and techniques of local mesh adaptation are discussed in the
following paragraphs.