Hierarchical Meshes and Local Multigrid Methods
C. Wieners (Uni Heidelberg)
We present a specification for hierarchically structured meshes which allows
the realization parallel local multigrid methods for general finite element
discretizations. For an efficient implementation of parallel multiplicative
multigrid methods it is required that the local defect evaluation and a local
smoothing procedure can be performed without data transfer between the mesh
levels. This is realized by additional copy elements on every levels in a
neighborhood of the refined elements.
We use the concept of vector classes for a precise definition of
copy elements which are required for a consistent defect computation on a
mesh hierarchy and a consistent defect restriction; furthermore, the smoothing
is required on the copy elements as well. Note that the definition of the
vector classes rely on the matrix graph and therefore on the discretization.
Moreover, we give a specification of the required geometry information for the
refinement process and for the implementation of higher order elements.
Local multigrid methods following these specifications are implemented
in the software system UG. We present examples for conforming,
nonconforming and mixed finite elements on hierarchically structured meshes.
This presentation is a joint work with the UG-group
and it generalizes the results in .
 P. Bastian, K. Birken, K. Johannsen, S. Lang, N. Neu\ss,
H. Rentz-Reichert, C. Wieners:
UG -- a flexible software toolbox for solving partial differential
Computing and Visualization in Science 1, (1997) 27-40