  The efficient generation of meshes is an important step in the numerical
solution of various problems in physics and engineering. We are interested in
situations where global mesh quality and tight coupling to the physical
solution is important. We consider elliptic PDE based mesh generation and
present a method for the construction of adaptive meshes in two spatial
dimensions using domain decomposition that is suitable for an implementation on
parallel computing architectures. The method uses the stochastic representation
of the exact solution of a linear mesh generator of Winslow type to find the
points of the adaptive mesh along the subdomain interfaces. The meshes over the
single subdomains can then be obtained completely independently of each other
using the probabilistically computed solutions along the interfaces as boundary
conditions for the linear mesh generator. Further to the previously
acknowledged performance characteristics, we demonstrate how the stochastic
domain decomposition approach is particularly suited to the problem of grid
generation - generating quality meshes efficiently. In addition we show further
improvements are possible using interpolation of the subdomain interfaces and
smoothing of mesh candidates. An optimal placement strategy is introduced to
automatically choose the number and placement of points along the interface
using the mesh density function. Various examples of meshes constructed using
this stochastic-deterministic domain decomposition technique are shown and
compared to the respective single domain solutions using a representative mesh
quality measure. A brief performance study is included to show the viability of
the stochastic domain decomposition approach and to illustrate the effect of
algorithmic choices on the solver's efficiency.
