Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 101

Sometimes, as in the following situation, it may be preferable to perform a
refinement on only part of a domain :
1. In some subdomain the derivatives (which determine the order of approxima-
tion) are much greater than in the rest of the domain. This may be clear from
the nature of the problem, or from the computation of error estimators which
will be dealt with in Ch. III, §7. In this case, refining this part of the grid can
lead to a reduction of the error in the entire domain.
2. We would like to start with a very coarse grid, and let the final grid be deter-
mined by automatic refinements. Often it is appropriate for the given problem
that the amount of refinement is different in different parts of the domain.
3. We want to compute the solution to higher accuracy in some subdomain.
The fact that in the ideal case it is even possible to carry out a refinement in
the direction of an edge or of a vertex using only similar triangles is illustrated in
Figs. 12 and 13. However, these are exceptional cases. Some care is required in
order to generate finer grids from coarser ones automatically. In particular, if more
than one level of refinement is used, we have to be careful to avoid thin triangles.
Fig. 26. Coarse grid (solid lines) and a refinement (dotted lines)
The following refinement rule , which can be found, e.g., in the multigrid algo-
rithm of Bank [1990], guarantees that each of the angles in the original triangulation
is bisected at most once. We may think of starting with a triangulation as in Fig. 26.
This triangulation contains several hanging nodes (cf. Fig. 11) which must be con-
verted to non-hanging nodes.
8.1 Refinement Rules.
(1) If an edge of a triangle T contains two or more vertices of other triangles (not
counting its own vertices), then the triangle T is divided into four congruent
subtriangles. This process is repeated until such triangles no longer exist.
(2) Every triangle which contains a vertex of another triangle at the midpoint of
one of its edges is divided into two parts. We call the new edge a green edge .