Parallel stochastic methods for PDE based grid generation

The efficient generation of meshes is an important step in the numerical solution of various problems in physics and engineering. Here we are interested in situations where global mesh quality and coupling to the physical solution is important. Hence we consider 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 sub-domain interfaces. The meshes over the single sub-domains 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 improvements through the use of interpolation along the sub-domain interfaces and smoothing of mesh candidates are discussed. 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.