subtask: Development of efficient solution techniques for Problems in domains
in which a simple triangulation is not constructable
Hier ist die deutsche Version dieser Seite.
Formulation of the task:
Including in the project A3 (fast solvers, analytical construction) of the
Sonderforschungsbereich 393 the following main task is considered.
Two hierarchically preconditioned methods (HPCCG) for the fast solution of
mesh equations that approximate two-dimensional elliptic boundary value
problems on arbitrary unstructured quasi uniform triangulations are developed,
cf. [Gl3,Gl7]. Based on the fictitious space approach the original problem can be
embedded into an auxiliary one, where both the hierarchical grid information
and the preconditioner by decomposing functions on it are well defined. We
implemented the corresponding Yserentant preconditioned conjugate gradient
method as well as the BPX-preconditioned cg-iteration having the optimal
computational costs, cf. [Gl7]. The designed software packages are described
also in [Gl7]. Several numerical examples demonstrate the efficiency of the
artificially constructed hierarchical methods which can be of enormous
importance in the industrial engineering, when often only the nodal coordinates
and the element connectivity of the underlying (fine) discretization are
available, see also [Gl4,Gl7]. The above strategy can be transfered to the computation
of real life threedimensional problems having the unstructured
discretization without hierarchical grid information, see [Gl8].
See this picture or this
postscript.
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H. Rhein und G. Globisch, October 1997