A Stochastic Domain Decomposition Method for Time Dependent Mesh Generation
We are interested in PDE based mesh generation. The mesh is computed as the solution of a mesh PDE which is coupled to the physical PDE of interest. In  we proposed a stochastic domain decomposition (SDD) method to find adaptive meshes for steady state problems by solving a linear elliptic mesh generator. The SDD approach, as originally formulated in , relies on a numerical evaluation of the probabilistic form of the exact solution of the linear elliptic boundary value problem. Monte-Carlo simulations are used to evaluate this probabilistic form only at the sub-domain interfaces. These interface approximations can be computed independently and are then used as Dirichlet boundary conditions for the deterministic sub-domain solves. It is generally not necessary to solve the mesh PDEs with high accuracy. Only a good quality mesh, one that allows an accurate representation of the physical PDE, is required. This lower accuracy requirement makes the proposed SDD method computationally more attractive, reducing the number of Monte-Carlo simulations required.
This research was supported by NSERC (Canada). AB is a recipient of an APART Fellowship of the Austrian Academy of Sciences. The authors thank Professor Weizhang Huang (Kansas) and the two anonymous referees for helpful remarks.