A new approach towards dynamic mesh adaptation is presented and used for transient flow
computations by the FEM. A gradient-based indicator is employed to identify elements which
need to be refined to achieve a prescribed accuracy. On the other hand, the mesh is coarsened
in overly resolved regions so as to reduce computational costs. Starting from an initial
triangulation T0 = (E0, V0) new cells are inserted following the red-green strategy by Bank et
al. [1], that is, elements which are marked for refinement are subdivided regularly. Hanging
nodes which are likely to occur in adjacent cells are eliminated by introducing transition
elements (green refinement). They are removed prior to performing further refinement in
order to prevent the gradual deterioration of the mesh quality due to acute angles. Each
subdivision of elements can be reversed by applying the corresponding inverse operation so
that both the refinement and the re-coarsening algorithm exhibit the same ‘throughput’.
For each vertex, its ‘date of birth’ is stored so that the maximum number of refinement
levels can be readily prescribed. Moreover, the entire genealogy of the sequence {Tm}M
m=0
of meshes can be reconstructed from the generation function g : Vm ! N0 which is defined
recursively based on the age of surrounding nodes. This idea dates back to the work
by Hempel [2], who proposed a special purpose adaptation procedure for triangular grids in
two dimensions. We extend his approach to arbitrary triangulations, whereby the characterization
of elements by means of generation numbers of corner nodes requires more involved
criteria. As a remedy, local two-level ordering is employed so that a priori knowledge about
inter-element relationships is available. In a practical implementation, binary arithmetic
can be adopted for the marking and identification of elements, whereby the association of
edges/vertices with individual bits relies on the consistent use of the local ordering strategy.
By construction, vertices with largest generation numbers have been inserted most recently,
and thus, they can be removed foremost. The node-locking algorithm [2] is reviewed
and a more general – yet simpler – version is proposed. The presented adaptation algorithms
are applied to convection-dominated problems and to compressible flows in 2D.